Differential Equations Answers

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example. (1) Equation cos(x+y) p+sin(x+y) q=z^2+z is a quasi -linear equation.

Find the integral surface of the partial differential equation

(X-y)y^2p+(y-x)x^2q=(x^2+y^2)z

Apply method of variations of parameters to solve the following differential equations

X^2 d^2y/dx^2 +xdy/dx -y=x^2 e^x

Solve the following differential equations changing the independent variables

X^2 d^2y/dx^2 - dy/dx - 4x^3 y= 8x^3 sinx^2 ,X>0.

Solve differential equations dy/dx=1+(dy/dx)^2.

(Xy^2 -x^2 )dx + ( 3x^2 y^2 + x^2 y-2x^3 + y^2) dy =0

Using the method of undetermined coefficients, write the trial solution of the equation

d^2y/dx^2 + 2dy/dx +5y= xe^-x cos2x


and hence solve it.

Solve the following equation :

(1) (D^2 - 2DD' + [D']^2 )z = tan(x - y)

(2) z(p - q) = z^2 + (x + y^2)

Find the value of n for which the equation (n - 1)^2 uxx - y^2n uyy = ny^2n-1 uy , is

(1) parabolic

(2) hyperbolic

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.

(1) Equation cos(x+y) p+sin(x+y) q=z^2+z is a quasi -linear equation.

(2) The solution of PDE dz/dx+dz/dy=z^2 is z=-[y+f(x-y) ].

(3) dz/dx.dz/dy-(dz/dy) ^2=0is a non -linear PDE.