Differential Equations Answers

d²z/dx²=sin y, solve it by 2 order integration

Question 1
1.1 Find the general solution of the system

dx/dt= -x + 4y + 2z
dy/dt= 4x - y - 2z
dz/dt= 6z

Solve the differential equation:
(x-y^2)dx+2xy.dy=0

The differential equation satisfied by a beam uniformly loaded at one end that is fixed and the other end 8s subjected to a tensile force P, is given by EI.d^2y/dx^2= P.y- 1/2(W.x^2),where E,I,P,W are constants. Show that the elastic curve for the beam under the conditions y=0, dy/dx=0 at x=0, is given by y(x)=(W/P^2)[1-cosh(vx)]+ (W/2P)(x^2+2/n^2),where EI= P/n^2

Using the separation of variable technique, solve: ∂u/∂x+ ∂u/∂y= 2(x+y)u

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

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