MA2020 Differential Equations (July - November 2014)Assignment Sheet- 1 (Covering Quiz-I Syllabus)

1. Solve:

(a) x dy

dx+ y = x3y6

(b) x dy

dx+ y = y2log(x)

(c) (x2y3 + xy) dydx

= 1

(d) dy

dx+ (2x tan−1 y − x3)(1 + y2) = 0

(e) (1 + ex

y )dx+ ex

y (1− xy)dy = 0

(f) (xdx+ ydy)(x2 + y2) = ydx− xdy

(g) dy

dx= 3y + 2e3x, y(0) = 2

(h) dy

dx= y tan(x) + sec(x), y(0) = −1

2. Define the Wronskian w(y1, y2) of any two differentiable functions y1 and y2defined in an interval (a, b) ⊂ R. Show that w(y1, y2) = 0 if y1 and y2 arelinearly dependent.

3. If y1 and y2 are any two solutions of a second order linear homogeneous ordinarydifferential equation which is defined in an interval (a, b) ⊂ R, then w(y1, y2) iseither identically zero or non-zero at any point of the interval (a, b).

4. If y1 and y2 are two linearly independent solutions of a second order linearhomogeneous ordinary differential equation then prove that y = c1y1 + c2y2,where c1 and c2 are constants, is a general solution.

5. Find the general solution of the following second order equations using the givenknown solution y1.

(a) x2 d2y

dx2 + x dy

dx− 4y = 0 where y1(x) = x2.

(b) (x− 1) d2y

dx2 − x dy

dx+ y = 0 where y1(x) = x.

(c) x d2y

dx2 − (2x+ 1) dydx

+ (x+ 1)y = 0 where y1(x) = ex.

6. Find the general solution of each of the following equations (Dn≡

dn

dxn )

(a) (D3− 4D2 + 5D − 2)y = 0

(b) (D2− 5D − 6)y = 3 sin 2x

(c) (D2− 4D + 4)y = cos 2x

(d) (D2− 3D + 2)y = (4x+ 5)e3x

(e) (D2− 1)y = 3e2x cos 2x

(f) (D2− 2D − 3)y = 3e−x cos x

7. Solve the following using the method of variation of parameters (Dn≡

dn

dxn )

(a) (D2 + 1)y = cosecx

(b) (D2−D − 6)y = e−x

1

(c) (D2 + a2)y = tan ax

(d) x2y′′ − 2xy′ + 2y = x3 cos x

8. Locate and classify the singular points of the following differential equaitons

(a) x2(x+ 2)y′′

+ xy′

− (2x− 1)y = 0

(b) (x− 1)2(x+ 3)y′′

+ (2x+ 1)y′

− y = 0

(c) (2x+ 1)x2y′′

− (x+ 2)y′

+ 2exy = 0

9. Solve, using the power series method

(a) (1− x2)y′′

− 2xy′

+ 2y = 0

(b) (1 + x2)y′′

+ 2xy′

− 2y = 0

(c) y′′

+ 1

xy

′

−1

x2y = 0, y(1) = 1, y′

(1) = 0

(d) y′′

+ x1−x2y

′

−1

1−x2y = 0, y(0) = 1, y′

(0) = 1

2