MTH102: ODE-Assignment-I

1.T Verify that y = 1/(x+ c) is general solution of y = y2. Find particular solutions suchthat (i) y(0) = 1, and (ii) y(0) = 1. In both the cases, find the largest interval I onwhich y is defined.

2.D Consider the differential equations y = y, x > 0, where is a constant. Show that

(i) if (x) is any solution and (x) = (x)ex, then (x) is a constant;

(ii) if < 0, then every solution tends to zero as x.

3.T Reduce the differential equation y = f(ax+ by +m

cx+ dy + n

), ad bc 6= 0 to a separable

form. Also discuss the case of ad = bc.

4.D Find general solution of the following differential equations:

(i) (x+ 2y + 1) (2x+ y 1)y = 0 (ii) y = (8x 2y + 1)2/(4x y 1)2

5.D Show that the following equations are exact and hence find their general solution:

(i) (cosx cos y cotx) = (sinx sin y)y (ii) y = 2x(yex2 y 3x)/(x2 + 3y2 + ex2)

6.T Show that the equation (3y2x) + 2y(y23x)y = 0 admits an integrating factor whichis a function of (x+ y2). Hence solve the differential equation.

7.D Show that 2 sin(y2) + xy cos(y2)y = 0 admits an integrating factor which is a function

of x only. Hence solve the differential equation.

8.T Reduce the following differential equations into linear form and solve:

(i) y2y + y3/x = sinx (ii) y sin y + x cos y = x (iii) y = y(xy3 1)

9.T Find the family of oblique trajectories which intersect the family of straight lines y = cx

at an angle of 45o.

10.D Show that the following families of curves are self-orthogonal:

(i) y2 = 4c(x+ c) (ii) x2/c2 + y2/(c2 1) = 1