1
MTL 102: Differential Equations
Problems set-2
1. Show that the following IVPs has unique solution for all a, b ∈ IR:
(a) x00 = x|x|, x(0) = a, x0 (0) = b,
(b) x00 = max{0, x|x|}, x(0) = a, x0 (0) = b
2. If f (x, t) be T −periodic with respect to t and let x(t) be a solution of x00 = f (t, x) defined
for all t ∈ IR, such that x(0) = x(T ) and x0 (0) = x0 (T ). Show that x(t) is periodic.
3. Let f (x) be differentiable function and let x : IR 7→ IR satisfy x(4) = f (x), x(i) (0) = 0 for
i = 1, 2, 3. Show that x(t) is even.
4. Find the solutions of the following initial value problems:
(a) y 00 − 2y 0 − 3y = 0, y(0) = 0, y 0 (0) = 1 (b) y 00 + 10y = 0, y(0) = π, y 0 (0) = π 2 .
5. Find a function φ which has a continuous derivative on 0 ≤ x ≤ 2 which satisfies φ(0) = 0,
φ0 (0) = 1, and y 00 − y = 0 for 0 ≤ x ≤ 1, and y 00 − 9y = 0 for 0 ≤ x ≤ 2.
6. Let φ1 , φ2 be two differentiable functions on an interval I, which are not necessarily
solutions of an equation L(y) = 0. Prove the following:
(a) If φ1 , φ2 are linearly dependent on I, then W (φ1 , φ2 )(x) = 0 for all x in I.
(b) If W (φ1 , φ2 )(x0 ) 6= 0 for some x0 in I, then φ1 , φ2 are linearly dependent on I.
(c) W (φ1 , φ2 )(x) = 0 for all x in I does not imply that φ1 , φ2 are linearly dependent
on I.
(d) W (φ1 , φ2 )(x) = 0 for all x in I, and φ2 (x) 6= 0 on I, imply that φ1 , φ2 are linearly
dependent on I.
7. Find all solutions of the following equations:
(a) 4y 00 − y = ex
(b) y 00 + 4y = cos x
(c) y 00 + 9y = sin 3x.
8. Let L(y) = y 00 + a1 y 0 + a2 y = 0, where a1 , a2 are constants, and let p be the characteristic
equation p(r) = r2 + a1 r + a2 .
(a) If A, α are constants and p(α) 6= 0, show that there is a solution φ of L(y) = Aeαx
of the form φ(x) = Beαx , where B is a constant.
(b) Compute a particular solution of L(y) = Aeαx in case p(α) = 0.
9. Are the following set of functions defined on −∞ < x < ∞ linearly dependent or independent there? Why?
(a) φ1 (x) = 1, φ2 (x) = x, φ3 (x) = x3
2
(c) φ1 (x) = x, φ2 (x) = e2x , φ3 (x) = |x|.
10. Use the method of undetermined coefficients to find a particular solution of each of the
following equations: (a) y 00 + 4y = cos x (b) y 00 + 4y = sin 2x (c) y 00 − y 0 − 2y = x2 + cos x
11. Find a real solution of the following Cauchy-Euler equations.
(a) x2 y 00 − 4xy 0 + 6y = 0, (b) 4x2 y 00 + 12xy 0 + 3y = 0, (c) x2 y 00 + 7xy 0 + 9y = 0,
(d) x2 y 00 − 2.5xy 0 − 2y = 0, (e) x2 y 00 + 7xy 0 + 13y = 0.
12. Solve the initial value problems.
(a) x2 y 00 − 2xy 0 + 2y = 0, y(1) = 1.5, y 0 (1) = 1.
(b) x2 y 00 + 3xy 0 + y = 0, y(1) = 3, y 0 (1) = −4.
(c) x2 y 00 − 3xy 0 + 4y = 0, y(1) = 0, y 0 (1) = 3.
13. Find all Linearly independent solutions of the following equations: (a) y 000 − 8y = 0
(b) y (4) + 16y = 0 (c) y (100) + 100y = 0 (d) y (4) − 16y = 0
14. Use the variation of parameters method to find a particular solution of the following
equations: (a) y 000 − y 0 = x, (b) y (4) + 16y = cos x, (c) y (4) − 4y (3) + 6y 00 − 4y 0 + y = ex .
15. Solve X 0 = AX, where


1 0 0


(a) A =  0 2 1 
1 0 3


 
1 0 1
1


 
(b) A =  0 −1 0  , X(0) =  0 
0 0 4
1
16. Solve the non-homogeneous system of equations
x0 = x + 2y + 2t
y 0 = 3y + t2
17. Solve the non-homogeneous system of equations
x0 = 2x + 6y + et
y 0 = x = 3y − et