Differential Equations Answers

Let f(x) = (x^2−1)/(x^4+1)

(a) At which points does the graph of the f(x) have a horizontal tangent line?

(b) Draw the graph of f(x) on MATLAB(or octave online) and identify the points for horizontal tangents on the graph.

Determine the general solution to the equation ∂²u/∂t²=c²(∂²u/∂x²) under the boundary conditions u(0,t)=u(1,t)=0 and initial conditions u(x,0)=Φ(x), u_t(x,o)=Ψ(x)

Find the characteristics values and the characteristics functions of the sturn_liouville problem

d/dx[x(dy/dx]+(λ /x)y=0,

y¹(1)=0, y¹(e(2π))=0 where λ>0

Use the laplace transform to solve the one dimensional boundary value problem

∂u/∂t=∂²u/∂x², 0<x<1, t>0

subject to u(x,0)=sin πx, u(0,t)=0, u(1,t)=0

Find the integral surface of the equation x²p+y²q+z²=0 passing through z=1,x+y=xy

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

∂u/∂x=4∂u/∂y, u(0,y)=8e^-3y

Find the non-trivial solution of the sturn_liouville problem d²y/dx²+λy=0 y(0)=0, y(π)=0

Define a partial differential equation and give an example.

Hence find the differential equation arising from F[y/x,z/x³]=0

Using the Laplace transform method, solve the partial differential equation

∂u/∂t=u-∂u/∂x. Subject to the initial condition u(x,0)=e^-7x, x>0,t>0