Differential Equations Answers

For the IVP, dy/dx= f(x,y), y(x0)= y0, the continuity of f(x,y) and ∂f/∂y guarantees the unique solution of the problem.
True or false with full explanation

Solve, using the method of variation of parameter,the following differential equation
d^2y/dx^2-2dy/dz=e^x.sinx

Find the general integral of the partial differential equation using Lagrange's method p(z+e^x)+q(z+e^y)= z^2-e^(x+y)

Problem:
x^2+y^2=9
How to find answer y''= -x^2+y^2/y^3=-9/y^3 ?( I can't describe the operation type therefore any solution is accepted)

Find the particular integral of the equation : (D^2-D')z=e^x+y

Solve (2D^2D'-3DD'^2+D'^3)z=0

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

Solve
(D^2+5DD'+5D'^2)z= x.sin(3x-2y)

Solve the following simultaneous differential equations
dx/y^2×(x-y)= dy/-x^2×(x-y)= dz/z×(x^2+y^2)

Using the transformation T=z^2/2, reduce the equation f(x,y,z,p,q)= x(y^2+z^2×q^2)- zy^2×p=0 to a form f(P,x)= g(Q,y) where p= ∂T/∂x, Q= ∂T/∂y