Question

Form Partial differential eq. of u=aeby+sinbxb

Answer 1

Answer on Question #68369 – Math – Differential Equations

Question

Form a partial differential equation of

$$u = a e^{by} + b \sin bx$$

Solution

If

$$u = a e^{by} + b \sin bx,$$

then

$$\frac{\partial u}{\partial x} = b^2 \cos bx,$$ $$\frac{\partial u}{\partial y} = a b e^{by},$$ $$\frac{\partial^2 u}{\partial x^2} = -b^3 \sin bx,$$ $$\frac{\partial^2 u}{\partial y^2} = a b^2 e^{by}.$$

Thus, the function

$$u = a e^{by} + b \sin bx$$

satisfies the partial differential equation

$$\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2} + b^2 u = 0.$$

On the other hand, the function

$$u = a e^{by} + b \sin bx$$

satisfies the system of partial differential equations:

$$\left\{ \begin{array}{l} \frac{\partial u}{\partial x} = b^2 \cos bx, \\ \frac{\partial u}{\partial y} = a b e^{by}. \end{array} \right.$$

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