Assignment Sheet 4
1. Find the general solution of the following second order equations using the given
known solution y1 .
d y 2 dy
(a) x2 dx 2 + x dx − y = 0 where y1 (x) = x.
d y 2 dy
(b) x2 dx 2
2 + x dx − 4y = 0 where y1 (x) = x .
d y 2 dy
(c) (x − 1) dx 2 − x dx + y = 0 where y1 (x) = x.
2
d y dy x
(d) x dx 2 − (2x + 1) dx + (x + 1)y = 0 where y1 (x) = e .
2
d y dy cos x
(e) x dx 2 + 2 dx + xy = 0 where y1 (x) = x
.
d y 2 dy
(f) x2 dx 3
2 − 5x dx + 9y = 0 where y1 (x) = x .
d y 2 dy 1
(g) 2x2 dx 2 + 3x dx − y = 0, x > 0 where y1 (x) = x .
2. Find the general solution of each of the following equations.
(a) y ′′ + 4y ′ + (π 2 + 4)y = 0.
(b) 9y ′′ − 30y ′ + 25y = 0.
3. Solve the following IVP.
(a) y ′′ + 4y ′ + (π 2 + 4)y = 0; y(1/2) = 1; y ′(1/2) = −2.
(b) 4y ′′ − 4y ′ − 3y = 0; y(−2) = e; y ′(−2) = −e/2.
4. Solve x2 y ′′ + xy ′ + y = 0; y(1) = 1; y ′(1) = 1.
5. Solve 4x2 y ′′ + 12xy ′ + 3y = 0; y(1) = 1; y ′(1) = 32 .
6. Find a second order homogeneous linear ODE for which the given functions are
solutions. Show linear independence by the Wronskian. Solve the IVP.
(a) x2 ; x2 lnx; y(1) = 4, y ′(1) = 6.
(b) e−kx cos πx, e−kx sin πx; y(0) = 1, y ′(0) = −k − π.
(c) 1, e−2x ; y(0) = 1, y ′(0) = −1.
7. Define the Wronskian w(y1 , y2) of any two differentiable functions y1 and y2
defined in an interval (a, b) ⊂ R. Show that w(y1 , y2) = 0 if y1 and y2 are
linearly dependent.
8. If y1 and y2 are any two solutions of a second order linear homogeneous ordinary
differential equation which is defined in an interval (a, b) ⊂ R, then w(y1, y2 ) is
either identically zero or non-zero at any point of the interval (a, b).
9. If y1 and y2 are two linearly independent solutions of a second order linear
homogeneous ordinary differential equation then prove that y = c1 y1 + c2 y2 ,
where c1 and c2 are constants, is a general solution.
1
dn
10. Find the general solution of each of the following equations (D n ≡ dxn
)
(a) (D 2 + 1)y = 2 cos x
(b) (D 2 − 3D + 2)y = (4x + 5)e3x
(c) (D 2 − 1)y = 3e2x cos 2x
(d) (D 2 − 2D + 3)y = 3e−x cos x
(e) (D 3 − 2D 2 − 5D + 6)y = 18ex
(f) (D 3 − 2D 2 + 4D − 8)y = 8(x2 + cos 2x)
(g) (D 3 + 3D 2 − 4)y = 12e−2x + 9ex
dn
11. Solve the following using the method of variation of parameters (D n ≡ dxn
)
(a) (D 2 + 1)y = cosecx
(b) (D 2 − D − 6)y = e−x
(c) (D 2 + a2 )y = tan ax
(d) (D 2 + a2 )y = sec 2x
(e) x2 y ′′ − 4xy ′ + 6y = 21x−4
(f) 4x2 y ′′ + 8xy ′ − 3y = 7x2 − 15x3
(g) x2 y ′′ − 2xy ′ + 2y = x3 cos x
(h) xy ′′ − y ′ = (3 + x)x2 ex
(i) (D 2 + 6D + 9)y = 16 xe2 +1
−3x
(j) y ′′ − 4y ′ + 4y = xe2x
(k) 2y ′′ − 3y ′ + y = (x2 + 1)ex
(l) y ′′ − 3y ′ + 2y = te3t + 1
(m) 3y ′′ + 4y ′ + y = sin x e−x ; y(0) = 1, y ′(0) = 0
12. Solve the following problems using method of undetermined coefficients (D n ≡
dn
dxn
)
(a) (D 2 + 25)y = 50 cos 5x + 30 sin 5x
(b) 8y ′′ − 6y ′ + y = 6 cosh x; y(0) = 15 ; y ′ (0) = 1
20
.
(c) (3D 2 + 27)y = 3 cos x + cos 3x
13. Show that the general solution of L(y) = f1 (x) + f2 (x), where L(y) is a linear
differential operator, is y = ȳ + y ∗ , where ȳ is the general solution of the
L(y) = 0 and y ∗ is the sum of the any particular solutions of L(y) = f1 (x) and
L(y) = f2 (x).
dn
14. Find the general solution of each of the following equations (D n ≡ dxn
)
(a) (D 4 − 81)y = 0
(b) (D 3 − 4D 2 + 5D − 2)y = 0
(c) (D 4 − 7D 3 + 18D 2 − 20D + 8)y = 0
(d) (D 4 + 2D 3 + 3D 2 + 2D + 1)y = 0
(e) (D 2 − 3D − 6)y = 3 sin 2x
