Question

Solve the differential equation \(5 \frac{∂u}{∂x}+3 \frac{∂u}{∂y}=2u \) using the method of separation of variables if \(u(0,y) = 9e^{-5y}.\)

(a) \(9e^{\frac{17}{5} x} e^{-5y} \)

(b) \(9e^{\frac{13}{5} x} e^{-5y} \)

(c) \(9e^{\frac{-17}{5} x} e^{-5y} \)

(d) \(9e^{\frac{-13}{5} x} e^{-5y} \)

The question was posed to me in semester exam.

The above asked question is from Solution of PDE by Variable Separation Method in section Applications of Partial Differential Equations of Engineering Mathematics

Answer 1

The correct option is (a) \(9e^{\frac{17}{5} x} e^{-5y} \)

The best I can explain: \( u(x,t) = X(x) T(t) \)

\(5X’Y + 3Y’X = 2XY \)

\(\frac{X’}{X} = \frac{k}{5} \) which implies \( X= c e^{\frac{k}{5} x} \)

\(\frac{Y’}{Y} = 2-k / 3\) which implies \( Y = c’ \, e^{\frac{2-k}{3} y} \)

Therefore \(u(x,t) = cc’ e^{\frac{k}{5} x} e^{\frac{2-k}{3} y} \)

\(u(0,y) = cc’ = e^{\frac{2-k}{3} y} \, 9 \, e^{-5y} \)

Hence cc’ = 9 and k = 17

Therefore, \(u(x,t) = 9 \, e^{\frac{17}{5} x} e^{-5y}. \)