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DE Tut4

This document is a tutorial sheet for the MA201 Differential Equation course at the Indian Institute of Technology Ropar for the 1st semester of the academic year 2018-19. It contains a series of problems related to second-order ordinary differential equations, including methods such as undetermined coefficients and variation of parameters. The problems cover finding particular solutions, general solutions, and specific cases related to Bessel's equation.

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13 views2 pages

DE Tut4

This document is a tutorial sheet for the MA201 Differential Equation course at the Indian Institute of Technology Ropar for the 1st semester of the academic year 2018-19. It contains a series of problems related to second-order ordinary differential equations, including methods such as undetermined coefficients and variation of parameters. The problems cover finding particular solutions, general solutions, and specific cases related to Bessel's equation.

Uploaded by

2023eeb1209
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Indian Institute of Technology Ropar

Department of Mathematics

MA201 : Differential Equation (UG Course)


1st semester of academic year 2018-19

Tutorial Sheet-4 Second Order ODE

1. Find particular solution of differential equation by method of undetermined coeffi-


cients:

(a) y 00 + 3y 0 = 2t4 + t2 e−3t + sin(3t)


(b) x2 y 00 − 2xy 0 + 2y = 6 ln(x)

2. If the number c is an m-fold root of polynomial equation P (r) = 0 then, P (r) =


Q(r)(r − c)m , where Q(c) 6= 0. Show that in this case the differential equation:

P (D)y = Aecx ,

possesses the solution


A
y(x) = xm ecx .
m!Q(c)
3. Find the solution by undetermined coefficients:

y 00 − y 0 − 2y = 4x2

that satisfies y(0) = 0 and y 0 (0) = 1.

4. Find particular solution of

y 00 − 3y 0 − 4y = 2e−t .

5. Determine the solution of


(
t, 0 ≤ t ≤ π,
y 00 + y =
πeπ−t , t > π,

satisfying the initial conditions y(0) = 0 and y 0 (0) = 1. Assume y and y 0 are
continuous at t = π.

6. Solve the ODE:


(
1, 0 ≤ t ≤ π/2,
y 00 + 2y 0 + 5y =
0, t > π/2,

with the initial conditions y(0) = 0 and y 0 (0) = 0.

7. Given y1 (t) = t−1 solution of

2t2 y 00 + 3ty 0 − y = 0, t > 0.

Find fundamental set of solutions.


8. Solve by method of variation of parameters:

(a) y 00 + 4y 0 + 4y = t−2 e−2t .


(b) y 00 + 4y = 8 tan(t), − π/2 < t < π/2.

9. Solve:
e2x
y 00 − 3y 0 + 2y = − .
ex + 1
10. The equation
1
x2 y 00 + xy 0 + (x2 − )y = 0, (a)
4
1
is a special case, corresponding to p = 2 of Bessel’s equation

x2 y 00 + xy 0 + (x2 − p2 )y = 0.

Verify that y1 (x) = x−1/2 is a solution of (a) for x > 0 and find general solution.

11. Find the general solution of the differential equation, given one solution of homoge-
neous equation:
xy 00 + (1 − 2x)y 0 + (x − 1)y = 0, y = ex .

12. Consider:
xy 00 − (1 + x)y 0 + y = x2 e2x .
Solve for general solution to this ODE.

13. Find the general solution of:

00 0 et
y − 2y + y = 2
+ 3et .
1+t

14. Solve:
y 00 + 4y = tan(2t) + e3t .

15. Solve the second-order non-homogeneous linear differential equation


√
x2 y 00 − 3xy + 4y = x,

over the interval (0, ∞).

16. Solve y 00 − y = ex by both the methods of undetermined coefficients and variation


of parameters. Explain any difference in the answers by two methods.

∗ ∗ ∗ ∗ ∗ ∗ ∗ End ∗ ∗ ∗ ∗ ∗ ∗ ∗

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