Question

Solve \(\frac{∂u}{∂x}=6 \frac{∂u}{∂t}+u\) using the method of separation of variables if u(x,0) = 10 e^-x.

(a) 10 e^-x e^-t/3

(b) 10 e^x e^-t/3

(c) 10 e^x/3 e^-t

(d) 10 e^-x/3 e^-t

I got this question in examination.

This intriguing question comes from Solution of PDE by Variable Separation Method topic in portion Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

The correct answer is (a) 10 e^-x e^-t/3

To explain: u(x,t) = X(x) T(t)

Substituting in the given equation, X’T = 6 T’X + XT

\(\frac{X’-X}{6X}=\frac{T’}{T}=k\)

\(\frac{X’}{X} = 1+6k \) which implies X = ce^(1+6k)x

\(\frac{T’}{T} = k \) which implies T = c’ e^kt

Therefore, u(x,t) = cc’ e^(1+6k)x e^kt

Now, u(x,0) = 10 e^-x = cc’e^(1+6k)x

Therefore, cc’ = 10 and  k = ^-1⁄3

Therefore, u(x,t) = 10 e^-x e^-t/3.