Differential Equations

Let y=y(x) be the solution of the differential equation 2x dx+ (e2y 2x+ x )dy=0, 0<x< 2, y( /4)=0. If y( /6)= , then e8 is equal to .

Let y=y(x) be the solution of the differential equation x dy+\2(1-x) x+x(2-x)\dx=0 such that y(0)=2. Then y(2) is equal to:

1. 2\ (2)+1\
2. 2
3. 1
4. 2\1- (2)\

Let y=y(x) be the solution of the differential equation (1-x2 )dy= [xy+ (x3+2 ) 3 (1-x2 ) ]dx, -1<x<1, y(0)=0. If y ( 12 )= mn, m and n are co-prime numbers, then m+n is equal to .

Let Y=Y(X) be a curve lying in the first quadrant such that the area enclosed by the line Y-y=Y (x)(X-x) and the co-ordinate axes, where (x, y) is any point on the curve, is always -y22 Y (x)+1, Y (x) 0. If Y(1)=1, then 12 Y(2) equals .

A function y=f(x) satisfies f(x) 2x+ x- (1+ 2x )f'(x)=0 with condition f(0)=0. Then, f ( 2 ) is equal to

1. 2
2. 1
3. -1
4. 0

If the solution curve y=y(x) of the differential equation (1+y2 ) (1+ ex )dx+x dy=0, x>0 passes through the point (1,1) and y(e)= - ( 32 ) + ( 32 ), then +2 is .

If ( yx )= e|x|+ 2 is the solution of the differential equation x ( yx ) dydx=y ( yx )+x and y(1)= 3, then 2 is equal to

1. 12
2. 9
3. 4
4. 3

Let f(x)= r x \ 2r2 [(f(r))2-f(x)f(r) ]r2-x2-r3e f(r)r \ be differentiable in (- , 0) (0, ) and f(1)=1. Then the value of ea, such that f(a)=0, is equal to .

Let x=x(t) and y=y(t) be solutions of the differential equations dxdt+ax=0 and dydt+by=0 respectively, a, b . Given that x(0)=2; y(0)=1 and 3y(1)=2x(1), the value of t, for which x(t)=y(t), is :

1. 232
2. 432
3. 4 3
4. 3 4

If the solution of the differential equation (2x+3y-2) dx+(4x+6y-7) dy=0, y(0)=3, is x+ y+3 e|2x+3y- |=6, then +2 +3 is equal to .