Differential Equations Answers

Given that u=u(x,y,z)=c₁,v=v(x,y,z)=c₂ are solutions of dx/P=dy/Q=dz/R. Show that F(u,v)=0 is a general solution of the Lagrange's equation

Pp+Qq=R

Determine the partial differential equation arising from

ax²+by²+z²=4

By the method of separation of variables solve the boundary value problem

∂u/∂x-5(∂u/∂t)=u given u(x,0)=e^-2x

Determine the general solution to the lagrange equation

y²zp-x²zq=x²y

use the method of separation of variables to determine the general solution to the one dimensional heat equation ∂u/∂t=α(∂²u/∂x²)=0,0≤ x≤ l,t≥ 0 subject to the initial condition u(x,0)=g(x). Hence compute the particular solution when g(x)=0

Find the characteristic values and the characteristic functions of the sturn_ liouville problem.

d/dx[x(dy/dx)]+(λ/x)y=0

y¹(1)=0,y¹(e^2x)=0 where λ is nonnegative

let F(u,v) where u=u(x,y,z) and v=v(x,y,z), further let z=z(x,y). show that on elimination of the arbitrary function F we obtain the first order partial differential equation;

Pp+Qq=R

Determine the laplace transform

i) ∂²u(x,t)/∂t²

ii)∂²u(x,t)/∂x²

Find the characteristic values and characteristic functions of the storm lionville problem

d²y/dx²+λy=0, y(0)=0,y(π)=0,

where λ is negative