Answer to Question #109553 in Differential Equations for David

Question

. Find the general solution of the partial differential equation

6uxx −5uxy −6uyy =52e3x+2y

Answer 1

The given equation can be written as

$(6D^{2}-5DD'-6D'^{2})u=52e^{3x+2y}$, where $D \equiv \dfrac{\partial}{\partial x}, D' \equiv \dfrac{\partial}{\partial y} .$

The auxiliary equation is

$6m^{2}-5m-6=0$ , solving we get

$m = -\dfrac{2}{3}, \dfrac{3}{2}$

$C.F=f_{1}(y-\frac{2x}{3})+f_{2}(y+\frac{3x}{2})$

$P.I = \dfrac{1}{6D^2-5DD'-6D'^{2}} 52e^{3x+2y}\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{(Replace $D$ by 3 and $D'$ by 2)}\\ = \dfrac{1}{6 \cdot 3^2-5\cdot 3\cdot 2-6\cdot2^{2}} 52e^{3x+2y}\\ = \dfrac{1}{0}52e^{3x+2y}~~\text{(Rule fails)}\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~\text{(Differentiating denominator with respect to $D$} \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{and multiplying the numerator by $x$)}\\ =x \dfrac{1}{12 D-5D'} 52e^{3x+2y}\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~\text{(Replace $D$ by 3 and $D'$ by 2)}\\ =x\dfrac{1}{36-10}52e^{3x+2y}\\ P.I=2xe^{3x+2y}$

The general solution is

$u(x,y) = C.F+P.I = f_{1}(y-\frac{2x}{3})+f_{2}(y+\frac{3x}{2})+2xe^{3x+2y}$