Sem3 tuto de

DEPARTMENT OF MATHEMATICAL SCIENCESFACULTY OF SCIENCE

UNIVERSITI TEKNOLOGI MALAYSIA

SSCM 1703 DIFFERENTIAL EQUATIONS TUTORIAL 1

1. Determine the longest interval in which the given initial value problem is certain tohave a unique solution.

(a) xy′′ + 3y = x; y(2) = 1, y′(2) = 3.

(b) (x− 1)y′′ − 3xy′ + 4y = sinx; y(−2) = 2, y′(−2) = 1.

(c) y′′ + (cosx)y′ + 3(lnx)y = 0; y(3) = 2, y′(3) = 1.

(d) (x− 2)y′′ + y′ + (x− 2)(tanx)y = 0; y(3) = 1, y′(3) = 2.

2. Determine the interval(s) of existence for the following differential equations.

(a) y(4) + 4y′′′ + 3y = x2. (b) xy′′′ + (sinx)y′′ + 3y = cosx.

(c) y′′′ + xy′′ + x2y′ + x3y = lnx. (d) x(x− 1)y(4) + exy′′ + 4x2y = 0.

3. Determine whether or not the following pairs of functions are linearly independentfor all x.

(a) sinx, cosx. (b) sinx cosx, sin 2x.

(c) e2x, xe2x. (d) sinh2 x, cosh2 x.

4. Determine whether the given set of functions is linearly dependent or linearly inde-pendent. If they are linearly dependent, find a linear relation among them.

(a) f1(x) = 2x− 3, f2(x) = x2 + 1, f3(x) = 2x2 − 3x.

(b) f1(x) = 2x− 3, f2(x) = 2x2 + 1, f3(x) = 3x2 + x.

(c) f1(x) = 2x− 3, f2(x) = x3 + 1, f3(x) = 2x2 − x, f4(x) = x2 + x+ 1.

(c) f1(x) = 2x− 3, f2(x) = x2 + 1, f3(x) = 2x2 − x, f4(x) = x2 + x+ 1.

5. Show that the given functions are a fundamental set of solutions by verifying theyare solutions of the differential equation and that their Wronskian is not zero. Thenform the general solution.

(a) y′′ + 4y = 0; y1(x) = cos 2x, y2(x) = sin 2x.

(b) (x− 1)y′′ − xy′ + y = 0; y1(x) = x, y2(x) = ex.

(c) y′′′ + 4y′ = 0; y1(x) = 1, y2(x) = cos 2x, y3(x) = sin 2x.

(d) xy′′′ − y′′ = 0; y1(x) = 1, y2(x) = x, y3(x) = x3.

6. Verify that the functions y1(x) = e−4x and y2(x) = e3x are solutions to the differentialequation y′′+y′−12y = 0. Then form the general solution. From this solution familyfind the member for which y(0) = 1, y′(0) = −2.

2

7. Show that the functions y1(x) = x3, y2(x) = x3 lnx and y3(x) = x−2 are solutions tox3y′′′ − x2y′′ − 6xy′ + 18y = 0. Use the Wronskian W [y1, y2, y3] to show that thesesolutions are linearly independent. Form the general solution to this third orderdifferential equation.

8. Show that any set of functions that contains the zero function is a linearly dependentset.

9. Show that y1(x) = 1 and y2(x) =√x are solutions of yy′′ + (y′)2 = 0, but that their

sum y = y1 + y2 is not a solution.

10. Show that if the functions y1(x) and y2(x) are solutions of y′′ + 2y2 = 0, it does notnecessarily follow that c1y1 + c2y2 is also a solution. Why?




1. Find the general solution for each of the following differential equations.

(a) y′′ − 4y′ + 3y = 0. (b) y′′ + 10y′ + 25y = 0.

(c) y′′ + 16y = 0. (d) 12y′′ − 5y′ − 2y = 0.

2. Find the particular solution for each of the following differential equations.

(a)d2y

dx2− 12

dy

dx+ 11y = 0, y(0) = 3, y′(0) = 11.

(b) 2d2y

dx2+ 5

dy

dx+ 2y = 0, y(0) = 0, y′(0) = 1.

(c)d2y

dx2− 4

dy

dx+ 4y = 0, y(1) = 4, y′(0) = 0.

(d)d2y

dx2− 6

dy

dx+ 13y = 0, y(π/2) = 0, y′(π/2) = 2.

3. The general solution of the equation

d2x

dt2+ ω2x = 0,

where ω is a constant, may be written in either of the forms

x = A cosωt+B sinωt = C sin (ωt+ α),

where A,B,C and α are constants.

(a) Determine the formula for C and α in terms of A and B.

(b) Taking ω = 6, find C and α given that x = 92

√3 and

dx

dt= 27 when t = 0.

4. Find the solution of the differential equation

6d2y

dx2+

dy

dx− 2y = 0,

which satisfy the conditions y′ = −4 when x = 0 and y → 0 as x → ∞.


d2y

dt2+ 6

dy

dt− 16y = 0,

which satisfy the conditions y(0) = 3 and y → 0 as t → ∞.





(a) y′′′ + y′′ + 4y′ + 4y = 0. (b) y′′′ − 4y′′ + 5y′ − 2y = 0.

(c) y(4) + 4y = 0. (d) y(4) − 5y′′ + 4y = 0.

2. Find the particular solution for each of the following differential equations.

(a)d3y

dx3+ 2

d2y

dx2+ 4

dy

dx= 0, y(0) = 0, y′(0) = 1, y′′(0) = 0.

(b)d3y

dx3+ 3

d2y

dx2+ 7

dy

dx+ 5y = 0, y(0) = 1, y′(0) = 0, y′′(0) = 0.

(c)d4y

dx4− d2y

dx2− 2y = 0, y(0) = 1, y′(0) = 0, y′′(0) = 0, y′′′(0) = 0.

(d)d4y

dx4− 2

d3y

dx3+

d2y

dx2− 2

dy

dx= 0, y(0) = 2, y′(0) = 0, y′′(0) = 1, y′′′(0) = 5.

3. Find the general solution

6d4y

dx4+ 5

d3y

dx3+ 25

d2y

dx2+ 20

dy

dx+ 4y = 0

given that y = sin 2x is one of the solution of the differential equation.

4. Find the particular solution of the differential equation

d3y

dx3+

d2y

dx2− dy

dx− y = 0

given that when x = 0, y = 1, when x = 2, y = 0 and also as x → ∞, y → 0.

5. Find the particular solution of the differential equation

y(4) + 6y′′′ + 9y′′ = 0

given that y = y′ = 0, y′′ = 6 when x = 0 and as x → ∞, y′ → 1. For this particularsolution, find the value of y when x = 1.

6. Find a differential equation whose general solution is:

(a) y = c1 e2t + c2 e

−3t.

(b) y = c1 + c2 cos 2t+ c3 sin 2t.




1. Verify that yp is a particular integral of the following differential equations.

(a) y′′ − 3y′ + 2y = x+ 4; yp = 14 (2x+ 11).

(b) y′′ + 9y = 3 cos 2x− sin 2x; yp = 15 (3 cos 2x− sin 2x).

2. Find the values of a, b, c if yp is a particular integral of the differential equations.

(a)d2y

dx2− dy

dx+ 12y = 2x; yp = ax+ b.

(b)d2y

dx2+ 6

dy

dx+ 8y = xe−4x; yp =

(ax2 + bx

)e−4x.

3. By using the method of the undetermined coefficients, find the general solution of the followingequations.

(a)d2y

dx2+ 5

dy

dx+ 4y = 3− 2x. (b)

d2z

dx2+ 4

dz

dx= 2x.

(c)d2z

dt2+ 2

dz

dt+ 3z = t− 2t2. (d) 2

d2y

dx2+ 5

dy

dx− 3y = 6x2 − 2x+ 1.

5. Solve the following Initial value problems (IVP’s).

(a)d2x

dt2+ x = 3− 2t2; x = 7 and

dx

dt= 0 when t = 0.

(b)d2y

dt2+

dy

dt+ 2y = 2t+ 4; y = 1 and

dy

dt= 2 when t = 0.


(a) y′′ − 2y′ − 8y = 3e4t; y(0) = 0, y′(0) = 3.

(b) z′′ − z′ − 6z = 3e−2x; z(0) = 0, z′(0) = 2.

7. By using the method of the undetermined method, find the general solution of the followingequations.

(a) y′′ − y′ − 2y = sin 2x. (b) y′′ + 4y = 12 cos 4x.

(c) y′′ − 4y′ + 3y = 2 cosx+ 4 sinx. (d) y′′ − 4y′ + 4y = −4 cos 2x.

8. Find the general solution of the following differential equations.

(a) y′′ + y′ − 2y = 2x− 40 cos 2x. (b) y′′ − 4y = 6 + e2x.

(c) y′′ − y′ − 2y = 6x+ 6e−x. (d) y′′ + 4y = sin 2x− cosx.

9. Find the general solution of the following differential equations.

(a) y′′ − y = (5x+ 1)e3x. (b) y′′ − 2y′ − 2y = x (ex − e−x).

(c) y′′ − y′ − 2y = ex cosx. (d) y′′ + 5y′ + 6y = e−2x sin 2x.

(e) y′′ + 4y′ + 4y = x2e−2x. (f) y′′ + 2y′ + y = x cosx.

10. Find the general solution of the differential equation

d2y

dx2+ 8

dy

dx+ 16y = x

(12− e−4x

).





(a) y′′′ + y = x3. (b) y′′′ − y′′ = x2 − 3x+ 1.

(c) y′′′ + 8y =(x2 + 1

)2. (d) y(4) − 8y′′ + 16y = 4x2 − 8x.

(e) y′′′ + 3y′′ + 2y′ = x2 + 2x. (f) y′′′ − y′′ − 6y′ = x2 + 1.


(a)d3y

dx3− 6

d2y

dx2+ 11

dy

dx− 6y = 6e−3x. (b)

d4y

dx4− 9

d2y

dx2= 12e−3x.

(c)d3y

dx3− 6

d2y

dx2+ 12

dy

dx− 8y = 21e2x. (d)

d4y

dx4− 16y = 4e2x.

(e)d4y

dx4− 2

d2y

dx2+ y = e2x + ex. (f)

d3y

dx3− 2

d2y

dx2+

dy

dx= 12e−x.


y(4) + 6y′′′ + 9y′′ = 9e−3x

which satisfies the initial conditions y = 2, y′ = 3, y′′ = y′′′ = 0 when x = 0.


(a) y′′′ − 3y′ + 2y = 4 sin 2x. (b) y(4) − 16y = 2 sin 2x.

(c) y′′′ − 3y′′ + 3y′ − y = cos 2x− 3 sin 2x. (d) y(4) + 8y′′ + 16y = 12 sin 4x.


(a)d3y

dx3+ 2

d2y

dx2= 2e−2x − 4e2x.

(b)d3y

dx3− 6

d2y

dx2+ 11

dy

dx− 6y = 3e2x − 5x+ 3x2.

(c)d4y

dx4+ 8

d2y

dx2+ 16y = 8x2 − 4 + 5 cos 3x− 15 sin 3x.


y′′′ − 2y′′ + y′ − 2y = 12 sin 2x− 4x

which satisfies the initial conditions y = 3, y′ = 5 and y′′ = −4 when x = 0.


d3y

dx3− 2

d2y

dx2− dy

dx+ 2y = 6x+ sin 2x

which satisfies the initial conditions y = 2,dy

dx= 3 and

d2y

dx2= 0 when x = 0.




1. Use variation of parameters to solve the following differential equations.

(a) y′′ − 4y′ + 3y = e3x. (b) y′′ + 4y = sin 2x.

(c) y′′ + y = sec3 x. (d) y′′ + 2y′ + 5y = e−x sin 2x.


(a) y′′ − 2y′ + y =ex

(x− 1)2. (b) y′′ + 2y′ + 2y = e−x secx.

(c) y′′ + y = ln cosx. (d) y′′ + y = sec2 x.


(a)d2y

dx2− y =

2

1 + ex. (b)

d2y

dx2+ y =

1

1 + sinx.

(c)d2y

dx2− 4

dy

dx+ 5y = e2x cscx. (d)

d2y

dx2− 3

dy

dx+ 2y =

e2x

1 + e2x.

4. Use variation of parameters and indicated solutions of the associated homogeneousequation, find a particular integral and then a general solution of each of the followingnonhomogeneous equations.

(a) x2y′′ − 2xy′ + 2y = x lnx, y1 = x, y2 = x2.

(b) xy′′ − (1 + x)y′ + y = x2e2x, y1 = ex, y2 = 1 + x.

(c) x2y′′ + xy′ − y =1

1 + x, y1 = x, y2 =

1

x.

5. Show that y = e−x and y = x− 1 are solutions of the homogeneous equation

xd2y

dx2+ (x− 1)

dy

dx− y = 0.

Hence, find the general solution of

xd2y

dx2+ (x− 1)

dy

dx− y =

x2

1− x, for 0 ≤ x < 1.

6. Show that y = ex and y = ex lnx are solutions of the homogeneous equation

xd2y

dx2− (2x− 1)

dy

dx+ (x− 1)y = 0.

Hence, use the method of variation of parameters to find the general solution of

xd2y

dx2− (2x− 1)

dy

dx+ (x− 1)y = xex, for x > 0.




1. Solve the following differential equations using the method of variation of parameters.

(a)d3y

dx3− 6

d2y

dx2+ 11

dy

dx− 6y = e3x.

(b)d3y

dx3− 3

d2y

dx2+ 3

dy

dx− y =

2ex

x2.

(c)d4y

dx4+ 2

d2y

dx2+ y = cos t.

2. Solve the following differential equations using the method of variation of parameters.

(a)d3y

dx3+

dy

dx= sec2 x.

(b)d3y

dx3+

dy

dx= 4 cotx.

(c)d3y

dx3+

dy

dx= tanx.

3. Find a formula involving integrals for a particular integral of the differential equation

y(4) − y = g(x).

Hint: The functions sin t, cos t, sinh t, and cosh t form a fundamental set of solutionsof the homogeneous equation.


y′′′ − 3y′′ + 3y′ − y = g(t).

If g(t) = t−2et, determine y(t).

5. Given that x, x2 and 1/x are solutions of the homogeneous equation corresponding to

x3y′′′ + x2y′′ − 2xy′ + 2y = 2x4, x > 0,

determine a particular integral.


x3y′′′ − 3x2y′′ + 6xy′ − 6y = g(x), x > 0.

Hint: Verify that x, x2 and x3 are solutions of the homogeneous equation.




1. A vibrating string without damping can be modeled by the differential equation

my′′ + ky = 0.

a. If m = 10 kg, k = 250 kg/sec2, y(0) = 0.3 m and y′(0) = −0.1 m/sec, find the equationof motion for this system.

b. When the equation of motion is of the form displayed in (a), the motion is said tobe oscillatory with frequency β/2π. Find the frequency of the oscillation for the springsystem of part (a).

2. A vibrating string with damping can be modelled by the differential equation

my′′ + by′ + ky = 0.

a. If m = 10 kg, k = 250 kg/sec2, b = 60 kg/sec, y(0) = 0.3 m and y′(0) = −0.1 m/sec, findthe equation of motion for this system.

b. Find the frequency of the oscillation.

c. Compare the results of this problem to Question 1 and determine what effect the dampinghas on the frequency of oscillation. What other effects does it have on the solution?

3. The motion of a certain mass-spring system with damping is governed by

y′′(t) + 6y′(t) + 16y(t) = 0

y(0) = 1, y′(0) = 0.

Find the equation of motion.

4. Determine the equation of motion for an undamped system at resonance governed by

d2y

dt2+ 9y = 2 cos 3t

y(0) = 1, y′(0) = 0.

Sketch the solution.

5. An undamped system is governed by

md2y

dt2+ ky = F0 cos γt;

y(0) = y′(0) = 0.

where γ = ω =√

km . Find the equation of motion of the system.

6. Consider the vibrations of a mass-spring system when a periodic force is applied. The systemis governed by the differential equation

mx′′ + bx′ + kx = F0 cos γt

where F0 and γ are nonnegative constants, and 0 < b2 < 4mk.

2

a. Show that the general solution to the corresponding homogeneous equation is

xh(t) = Ae(−b/m)t sin

(√4mk − b2

2mt+ ϕ

).

b. Show that the general solution to the nonhomogeneous problem is given by

x(t) = Ae(−b/m)t sin

(√4mk − b2

2mt+ ϕ

)+

F0√(k −mγ2)2 + b2γ2

sin(γt+ θ).

7. RLC Series Circuit. In the study of an electrical circuit consisting of a resistor, capacitor,inductor, and an electromotive force, we are led to an initial value problem of the form

LdI

dt+RI +

q

C= E(t) (1)

q(0) = q0, I(0) = I0.

where L is the inductance in henrys, R is the resistance in ohms, C is the capacitance infarads, E(t) is the electromotive force in volts, q(t) is the charge in coulombs on the capacitor

at time t, and I =dq

dtis the current in amperes.

Find the current at time t if the charge on the capacitor is initially zero, the initial current is0, L = 10 henrys, R = 20 ohms, C = 6260−1 farads and E(t) = 100 volts.

Hint: Differentiate both sides of the differential equation to obtain a homogeneous linearsecond order equation for I(t). Then use equation (1) to determine dI

dt at t = 0.

8. An RLC series circuit has an elctromotive force given by E(t) = sin 100t volts, a resistor of 0.02ohms, an inductor of 0.001 henrys, and a capacitor of 2 farads. If the initial current and theinitial charge on the capacitor are both zero, determine the current in the circuit for t > 0.

SOLUTIONS TO TUTORIAL 8

1. a. y = 0.3 cos 5t− 0.02 sin 5t. b. 0.8Hz

2. a. y = e−3t(0.3 cos 4t+ 0.2 sin 4t). b. 5/2π.

c. Exponential factor (damping factor)

3. y = e−3x

(cos

√7x+

3√7sin

√7x

).

4. y = cos 3t+1

2t sin t.

5. y =−F0

m(ω2 − γ2)cosωt+

F0

m(ω2 − γ2)sin γt.

7. I =2

5e−t sin 25t.

8. I = e−10t

(95

9.425cos 20t− 105

18.85sin 20t

)− 95

9.425cos 100t+

20

9.425sin 100t.




1. Use the method of reduction of order to find the general solution to the homoge-neous differential equation using the given y1(x). Then, state the second linearlyindependent solution.

(a) y′′ − 5y′ + 6y = 0; y1 = e2x.

(b) 4y′′ + 12y′ + 9y = 0; y1 = e−32x.

(c) x2y′′ + 2xy′ − 2y = 0, x > 0; y1 = x.

(d) x2y′′ + 3xy′ + y = 0, x > 0; y1 = x−1.

(e) (x− 1)y′′ − xy′ + y = 0, x > 1; y1 = ex.

2. Use the method of reduction of order to find the general solution to the nonhomo-geneous differential equation using the given y1(x). Then, state the second linearlyindependent solution and the particular integral.

(a) y′′ − 4y′ + 3y = 9e2x, y1 = e3x.

(b) y′′ − 6y′ + 8y = e4x, y1 = e2x.

(c) x2y′′ + xy′ − y =√x, y1 = x.

(d) xy′′ + (2 + 2x)y′ + 2y = 8e2x, y1 = x−1.

(e) (x+ 1)y′′ + xy′ − y = (x+ 1)2, y1 = e−x.

3. Show that y = e−x is a solution of the homogeneous equation

xd2y

dx2+ (x− 1)

dy

dx− y = 0.

Hence, find the general solution of

xd2y

dx2+ (x− 1)

dy

dx− y =

x2

1− x, for 0 ≤ x < 1.




1. Show that if x = et, then

xdy

dx=

dy

dtand x2

d2y

dx2=

d2y

dt2− dy

dt.

Hence, solve the following differential equations.

(a) x2d2y

dx2+ 3x

dy

dx+ 2y = 0. (b) x

d2y

dx2− 2

dy

dx= 0.

(c) x2d2y

dx2+ 6x

dy

dx+ 4y = 2 lnx. (d) x2

d2y

dx2+ 4x

dy

dx+ 2y = x− 1

x.

2. Solve the following differential equations.

(a) x2d2y

dx2− 5x

dy

dx+ 8y = 0. (b) x2

d2y

dx2+ 3x

dy

dx+ 2y = 0.

(c) x2d2y

dx2− x

dy

dx+ y = x3. (d) x2

d2y

dx2+ 5x

dy

dx+ 5y = 2x+ x2.

(e) 9x2d2y

dx2+ 3x

dy

dx+ y = x lnx. (f) x2

d2y

dx2− 2x

dy

dx− 4y = x2 + 2 lnx.


(a) x2d2y

dx2− 10x

dy

dx+ 24y = 6x2; y(1) = 1, y′(1) = 1.

(b) x2d2y

dx2+ 3x

dy

dx+ y =

1

x; y(1) = 2, y′(1) = 1.

(c) 4x2d2y

dx2− 3y = 4x2; y(1) = 0, y′(1) = 0.

(d) x2d2y

dx2− 2x

dy

dx+ 2y = (x− 1) lnx; y(1) = 0, y′(1) = −1.

(e) x2d2y

dx2− 3x

dy

dx+ 4y = 6x2 lnx+

6

x; y(1) = 0, y′(1) = 1.

(f) x2d2y

dx2− 2x

dy

dx+ 2y = 4x+ sin (lnx); y(1) = 1, y′(1) = 0.

4. Find a general solution for Cauchy-Euler’s differential equation

x2d2y

dx2+ x

dy

dx= 12 lnx.

5. Find the general solution of

x2d2y

dx2− 3x

dy

dx+ 4y = x2 (lnx+ 1) , for x > 0.




1. Which of the following functions are of exponential order?

(a) f(t) = t2. (b) f(t) = 5e−t. (c) f(t) = sin 2t.

(d) f(t) = tan t. (e) f(t) = 1/t. (f) f(t) = t2e2t.

2. Use the definition of the Laplace transform to find L{f(t)} for the given f(t).

(a) f(t) = e3t. (b) f(t) = tn, n = 1, 2, 3, . . . .

(c) f(t) =

{0, 0 ≤ t < 4

1, t ≥ 4(d) f(t) =

{1, 0 ≤ t < 1

t, t ≥ 1

(e) f(t) =

et, 0 < t < 2

0, 2 < t < 4

5, t ≥ 4

(f) f(t) =

3, 0 ≤ t < 3

t, 3 ≤ t < 5

e−2t, t ≥ 5

3. Prove that if c1 and c2 are any constants while f1(t) and f2(t) are functions with Laplacetranforms F1(s) and F2(s) respectively, then

L{c1f1(t) + c2f2(t)} = c1F1(s) + c2F2(s).

Use this result to find the Laplace transform of the following functions.

(a) f(t) = 2 sin 3t+ 3 cos 2t. (b) f(t) =(2t− e−2t

)2.

(c) f(t) = e−2t + sinh 2t. (d) f(t) = cosh2 2t.

(e) f(t) = cos2 3t. (f) f(t) = (1 + sin t)2.

4. Show that if L{f(t)} = F (s) and a is a constant, then

L{eat f(t)} = F (s− a).


(a) f(t) = t2e3t. (b) f(t) = e−2t sin 3t.

(c) f(t) = e5t cos 3t. (d) f(t) = e−4t cosh 4t.

5. If L{f(t)} = F (s), then for n = 1, 2, 3, . . .

L{tn f(t)} = (−1)ndn

dsnF (s).


(a) f(t) = t2e3t. (b) f(t) = t cos 2t.

(c) f(t) = t cos(4t− 1

3π). (d) f(t) = t2 cosh 4t.

2

6. Use the definition of the Laplace transform to find L{f(t)} if

f(t) =

{1− t, 0 ≤ t < 1

0, t ≥ 1.

Hence, evaluate

(a) L{etf(t)}. (b) L{tf(t)}.

7. Theorem: If L{f(t)} = F (s) and limt→0

f(t)

texists, then

L{f(t)

t

}=

∫ ∞

s

F (u)du


(a)e2t − 1

t. (b)

1− cos 3t

t.

(c)e−3t sin 2t

t. (d)

e−at − e−bt

t.

8. Theorem: If L{f(t)} = F (s), then

L{∫ t

0

f(u) du

}=

F (s)

s.


(a)

∫ t

0

cosu du. (b) t

∫ t

0

cosu du.

(c)

∫ t

0

1− e−u

udu. (d)

∫ t

0

eu − cos 2u

udu.

9. Find the Laplace transform of each of the following functions.

(a) H(t− 2). (b) cos (t− 1)H(t− 1).

(c) (2t− 5)H(t− 3). (d)(t2 − 1

)H(t− 2).

(e) e2t H(t− 2). (f) (t− sin (2t− 12π))H(t− 1

2π).

10. Sketch the graph of the following functions. Express in terms of unit step functions. Hencefind its Laplace transform.

(a) f(t) =

{e−2t 0 ≤ t < 2

2 t ≥ 2.(b) f(t) =

{sin t 0 ≤ t < π

0 t ≥ π.

(c) f(t) =

et 0 < t < 2

0 2 < t < 4

5 t > 4.

(d) f(t) =

3 0 ≤ t < 3

t 3 ≤ t < 5

e−2t t ≥ 5.

11. Evaluate

(a) L{2 δ(t− 1)}. (b) L{t2 δ(t− 3)}.

(c) L{e−4t δ(t− 1)}. (d) L{(8t− 5) δ(t− 3)}.

(e) L{(sin 2t) δ(t− 12π)}. (f) L{cosh (5t+ 2) δ(t− 2)}.




1. Find the inverse of each of the following transforms.

(a)2

s3. (b)

3

s5. (c)

s+ 3

(s+ 1)(s− 3).

(d)1

(s+ 2)4. (e)

s

(s+ 2)4. (f)

2− s

(s− 3)4.

(g)s+ 1

9s2 + 12s+ 4. (h)

1

s(s+ 2)2. (i)

1

s2(s+ 1).

2. Find the inverse of each of the following transforms.

(a)6

(s− 1)2 + 4. (b)

s+ 3

(s+ 3)2 + 9. (c)

5s+ 3

s2 + 4s− 5.

(d)2s− 3

2s2 − 8s+ 16. (e)

3s+ 2

s2 + 2s+ 10. (f)

5− s

s2 + 2s+ 2.

(g)e−3s

s2 − 9. (h)

e−s

(s+ 1)3. (i)

e−s + e−2s

s2 − 3s+ 2.

3. If f(t) is a continuous function for t ≥ 0 and

f(t) = L−1

{3 (1 + e−πs)

s2 + 9

},

evaluate f( 12π), f(π) and f(2π).

4. If f(t) is a continuous function for t ≥ 0 and

f(t) = L−1

{e−4s

(s+ 2)3

},

evaluate f(2), f(5) and f(7).

5. By using the convolution theorem, find the inverse of the following transforms.

(a)1

(s+ 1)(s− 2). (b)

1

s(s+ 2)2.

(c)1

s2(s− 1)2. (d)

1

(s+ 1)(s2 + 1).

(e)s

(s2 + 1)2 . (f)

s2

(s2 + 1)2 .


d2y

dt2+ 4

dy

dt+ 13y =

1

3e−2t sin 3t

for which y = 1 and y′ = −2 when t = 0.

7. Find the solution tod2y

dt2+ 2

dy

dt+ y = H(t)−H(t− 1),

satisfying y(0) = 1 and y′(0) = 0, where H(t) is the Heaviside unit function.

2

8. By using convolution theorem, show that

L−1

{1

(s2 + a2)2

}=

1

2a3(sin at− at cos at).

Hence solve the differential equation

d2y

dt2+ y = t cos 2t

given that y anddy

dtare both zero when t = 0.

9. (a) Use the convolution theorem to show that

L−1

{s

(s2 + 1)2

}=

1

2t sin t.

(b) Use the result in (a), to solve the initial value problem

d2y

dt2+ y = cos t− (cos t)H(t− π)

given that y(0) = 2 and y′(0) = 1.

10. (a) Prove that

L{1

2sin 2t− t cos 2t

}=

8

(s2 + 4)2 .

(b) A damped forced vibration is given by

d2y

dt2+ 2

dy

dt+ 5y = e−t sin 2t,

which satisfies the initial conditions y(0) = 0 and y′(0) = 1.By using Laplace transform find y(t). Hence, show that, it t is small enough,

y(t) ≈ t− t2.

11. (a) Express1

(s+ 1)(s+ 2)2in partial fractions and show that

L−1

{1

(s+ 1)(s+ 2)2

}= e−t − (1 + t)e−2t.

(b) Use the result in part (a) to solve the initial value problem

d2y

dt2+ 4

dy

dt+ 4y = f(t), y(0) = 1, y′(0) = −1

where

f(t) =

{0 0 ≤ t < 2e−(t−2) t ≥ 2.

12. The displacement x of a forced system at time t is described by the differential equation

d2x

dt2+ 3

dx

dt+ 2x = sin t+ δ(t− 1),

where δ(t), the Dirac delta function, models the effect of a unit impulsive force at timet = 0. Given x(0) = − 3

10 and x′(0) = 110 , use Laplace transform to find x at any time t.

3

13. By using the Laplace transform solve the differential equation

d2x

dt2+ 4

dx

dt+ 3x = 2 δ(t− 1)

which satisfies the conditions x(0) = 0 and x′(0) = 3 where δ(t − 1) is a delta function.Find x(2) and also the value of x when x → ∞.

14. Solve the following systems of simultaneous differential equations using Laplace transform:

(a)dx

dt+ y = sin t,

dy

dt+ x = cos t, given that x = 2, y = 0 when t = 0.

(b) 3dx

dt+

dy

dt+ 2x = 1,

dx

dt+ 4

dy

dt+ 3y = 0, given that x(0) = 3 and y(0) = 0.

15. The current i1 and i2 are given by the differential equations

di1dt

− ωi2 = a cos pt

di2dt

+ ωi1 = a sin pt,

where a, p, ω are constants and p = ω. Find the current i1 and i2 if i1 = i2 = 0 when t = 0.

16. Solve the system of the simultaneous differential equations

dx

dt+ y = et, x(0) = 1

x− dy

dt= t, y(0) = 1.

17. Solve the system of the simultaneous differential equations

dx

dt+ 2x =

dy

dt+ 10 cos t,

dy

dt+ 2y = 4e−2t − dx

dt,

given that x = 2, y = 0 when t = 0.

18. Given that x and y are functions of t such that x = 3x− y, y = x+ y, and x = 1, y = 0 att = 0, show that x− y = e2t.

19. A point (x, y) moves in accordance with the equations

x+ 2y = 5et, y − 2x = 5et.

It is given that x = −1 and y = 3 when t = 0. Show that the point moves in a straight line.

20. The coordinates of a point P moving in the xy−plane satisfy the differential equations

x+ y = sin 2t, y − x = 2 cos 2t.

It is given that x = 1 and y = 0 when t = 0. Prove that the path of the point is given by

y2 =(1− x2

)(2x+ 1)2.

Note For brevity, x, y, are written fordx

dt,dy

dtrespectively.

UNIVERSITI TEKNOLOGI MALAYSIA FACULTY OF SCIENCE

DEPARTMENT OF SCIENCE MATHEMATICS

TIME : 75 Minutes TEST 2 (25%) SESSION/SEM: 20112012/02 Answer all questions. 1. Show that the substitution transforms the Cauchy Euler’s differential

equation

to

Hence, find the general solution for in terms of (10 marks)

2. Solve the following initial value problems

(7 marks)

3. Given

( ) {

.

a) Sketch the graph of ( ) (2 marks) b) Use the definition of the Laplace Transform to find * ( )+. (6 marks)

4. a) Evaluate * ( ) + (5 marks) b) Express ( ) in terms of unit step function and hence find its Laplace Transform.

( ) {

(6 marks)

5. Find the inverse Laplace transform of the following functions:

)

)

( )( )

6. If ( ) is a continuous function for and

( ) { ( )

( ) }

evaluate (

), ( ) and ( ) (6 marks)

(4 marks)

(4 marks)

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