Answer to Question #225311 in Differential Equations for Anuj

Question

Solve the method of separation of variables du/dx+u=du/dt if u=4e–3x when t= 0

Answer 1

Solution

Let u(x,t) = X(x)T(t)

Substituting u back into equation and dividing by X(x)T(t)

$\frac{X'}{X}+1=\frac{T'}{T}$

Since the right hand side depends only on x and the left hand side only on t, both sides are equal to some constant value −C. Thus,

$\frac{X'}{X}+1=-C,\ \ \ \ \frac{T'}{T}=-C$

From this ODE: X(x) = Ae^-(C+1)x, T(t) = Be^-Ct  , u(x,t) = De^-(C+1)x-Ct (A,B,C,D  are arbitrary constants)

From initial condition u(x,0)=4e^–3x  =>  De^-(C+1) = 4e^–3x  =>  D=4, C=2

Therefore the solution is u(x,t) = 4e^-3x-2t