Answer to Question #294429 in Differential Equations for Lucky

Question

How do you solve the partial differential equation (3D^² +10DD^1+3D ^1^2) z = e^x-y?

Answer 1

we shall solve the partial differential equation

$(3D^{2}+10DD^{\prime}+3D^{\prime^{2}})z=e^{x-y}$

The auxilliary equation is

$3m^{2}+10m+3=0\\(3m+1)(m+3)=0$

$3m+1=0$ or $m+3=0$

$m=-\frac{1}{3},3$

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

We have the P.I to be

$P.I=\frac{1}{3D^{2}+10DD^{\prime}+3D^{\prime^{2}}}e^{x-y}$

$=\frac{1}{3(1)^{2}+10(1)(-1)+3(-1)^{2}}e^{x-y}$

$=\frac{1}{3-10+3}e^{x-y}$

$=-\frac{1}{4}e^{x-y}$

$z=f_{1}(y-\frac{1}{3}x)+f_{2}(y-3x)-\frac{1}{4}e^{x-y}$

which is the general solution for the differential equation.