Question

What is the solution of, \(\frac{∂^2 u}{∂x^2}=2xe^t,\) after applying method of separation of variables \((u(0,t)=t,\frac{∂u}{∂x} (0,t)= e^t)\)?

(a) \(u=\frac{x^3}{3} e^t+xe^t \)

(b) \(u=\frac{x^3}{3} e^t+xe^t+t\)

(c) \(u=\frac{x^3}{3} e^t+e^t+t\)

(d) \(u=\frac{x^2}{2} e^t+xe^t+t\)

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This question is from Method of Separation of Variables in chapter Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

Right option is (b) \(u=\frac{x^3}{3} e^t+xe^t+t\)

Easy explanation: Given: \(\frac{∂^2 u}{∂x^2}=2xe^t,\)……………………………………………………………………………… (1)

Integrating (1) with respect to x we get,

\(\frac{∂u}{∂x}=x^2 e^t+f(t),\) where f(t) = arbitrary function ………………………………. (2)

Integrating again with respect to x we get,

\(u=\frac{x^3}{3} e^t+xf(t)+g(t)\)………………………………………………………………………………………… (3)

Applying the given initial condition, \(\frac{∂u}{∂x}(0,t)= e^t\) in equation (2), we get, f(t)=e^t.

Applying the initial condition, u(0,t)=t in equation (3), we get, g(t)=t.

Therefore, the solution, \(u=\frac{x^3}{3} e^t+xe^t+t.\)