Question

Show that the following equation has a solution of the form u(x,y) = e^(ax+by) and find the constants a and b:

U_xx + U_yy = 5e^(x-2y).

Answer 1

Show that the following equation has a solution of the form $u(x,y) = e^{(ax + by)}$ and find the constants $a$ and $b$ :

$$u _ {x x} + u _ {y y} = 5 e ^ {(x - 2 y)}$$

**Solution:**

$$u _ {x} = \frac {\partial u}{\partial x} = a e ^ {(a x + b y)}$$ $$u _ {x x} = \frac {\partial^ {2} u}{\partial x ^ {2}} = a ^ {2} e ^ {(a x + b y)}$$ $$u _ {y} = \frac {\partial u}{\partial y} = b e ^ {(a x + b y)}$$ $$u _ {y y} = \frac {\partial^ {2} u}{\partial y ^ {2}} = b ^ {2} e ^ {(a x + b y)}$$ $$u _ {x x} + u _ {y y} = a ^ {2} e ^ {(a x + b y)} + b ^ {2} e ^ {(a x + b y)} = (a ^ {2} + b ^ {2}) e ^ {(a x + b y)}$$ $$(a ^ {2} + b ^ {2}) e ^ {(a x + b y)} = 5 e ^ {(x - 2 y)} \rightarrow a = 1, b = - 2$$

Answer: $a = 1, b = -2$ .