Differential Equations Answers

show that the Fourier series of
f(x)=e^x (-π,π)
Is
1/π sin hπ+∑_(n=1)^∞▒〖(2 sin hπ)/π(1+n^2 ) (-1)^n (cos nx-nsin nx)〗

Define the laplace equation in 2d and 3d also comment regarding the classification of Laplace equation and its application?

Solve y zp + x²zq = y2x

(1+yz)dx + x(z-x)dy -(1+xy)dz=0

.Find the Fourier sine and cosine integral for the following functions
i. f(x)={█(x^2,0<x≤10@0,x>10)┤ ii. f(x)={█(x,0≤x≤1@x+1,1<x<2@0,x≥2)┤

λλA particle falls from rest in a medium in which the resistance is λv^2 per unit mass, v being velocity of the particle at time t. Prove that the distance fallen in time to is 1/λ[Incos(t(√gλ))],where g is the acceleration due to gravity

Solve
dx/y^2+yz+z^2=dy/z^2+zx+x^2=dz/x^2+xy+y^2

Using the method of vibration of parameters,solve the equation
d^2y/dx^2 +a^2y =secax

Solve
p^2 -2xyp+4y^2=0,where p=dy/dx