Solve-the-PDE-by-method-of-separating-variables-2-u-x-2-2t-2-u-x-t-4u-0-

Question Number 81937 by Joel578 last updated on 16/Feb/20

Solve the PDE by method of separating variables

\frac{\partial^{2} u}{\partial x^{2}} + 2 t \frac{\partial^{2} u}{\partial x \partial t} - 4 u = 0

Commented by Joel578 last updated on 16/Feb/20

My approach

Let the solution be u (x, t) = F (x) G (t)

Substitute u (x, t) to original eq . and divide

by F (x) G (t) yields

\frac{1}{F} \frac{d^{2} F}{{d x}^{2}} + 2 (\frac{1}{F} \frac{d F}{d x}) (\frac{t}{T} \frac{d T}{d t}) - 4 = 0

Let P (x) = \frac{1}{F} \frac{d^{2} F}{{d x}^{2}}, M (x) = \frac{2}{F} \frac{d F}{d x}, N (t) = \frac{t}{T} \frac{d T}{d t}

\Rightarrow P (x) + M (x) N (t) - 4 = 0

Differentiate both sides with respect to x, then t

\Rightarrow M^{'} (x) N^{'} (t) = 0

which means either M (x) or N (t) is a constant

∙ Case 1 : N (t) = n

P (x) + n M (x) - 4 = 0

\Rightarrow \frac{d^{2} F}{{d x}^{2}} + 2 n \frac{d F}{d x} - 4 F = 0

\Rightarrow F (x) = C_{1} e^{(- n + 2 \sqrt{4 + n^{2}}) x} + C_{2} e^{(- n - 2 \sqrt{4 + n^{2}}) x}

\Rightarrow N (t) = \frac{t}{T} \frac{d T}{d t} = n \Rightarrow T (t) = C_{3} t^{n}

∙ Case 2 : M (x) = m

P (x) + m N (t) - 4 = 0

Commented by Joel578 last updated on 16/Feb/20

P l e a s e g u i d e m e w i t h t h e s e c o n d c a s e .

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