Differential Equations Answers

Solve, using the method of variation of parameters
d^2y/dx^2-y=2/1+e^x

a) Solve the differential equation
dy/dx+(x/1-x^2)y=x (sqrt y),y(0)=1

Given that
y( x )= x^-1 is one solution of the differential equation
2x^2y''+3xy'-y=0,x>0,
find a second linearly independent solution of the equation

a) Solve the differential equation
dy/dx+xy=y^2e^(x^2/2)sinx

b) Find the solution of the Riccati equation
dy/dx=(2cos^2x-sin^2x+y^2)/2cosx ;y(x)=sinx

Find the value of b for which the equation
(1 )( ye^2xy + x )dx + bxe^2xy dy = 0
is exact, and hence solve it for that value of b

State whether the following statements are true or false. Justify yourself with the help of
a short proof or a counter example(1)Equation x^2 ( y − px) = yp^2 is reducible to clairaut’s form

State whether the following statements are true or false. Justify yourself with the help of
a short proof or a counter example(1).y = 0, is a singular solution of the differential equation 27y-8(dy/dx)^3=0

State whether the following statements are true or false. Justify yourself with the help of
a short proof or a counter example.
i) y'+ P(x) y = Q(x) y^2 is a linear equation for integral values of n.

State whether the following statements are true or false. Give reasons for your answers (2) The function f(x, y) =max{y/x, x} is a homogenous function on R^2