Question 1: Solve the following Homogeneous Linear PDEs with constant Coefficients
\[3\frac{\partial^2z}{\partial x^2}+2\frac{\partial^2z}{\partial x\partial y}-5\frac{\partial^2z}{\partial y^2}=0\]
Solution :
Given PDE
\[3\frac{\partial^2z}{\partial x^2}+2\frac{\partial^2z}{\partial x\partial y}-5\frac{\partial^2z}{\partial y^2}=0 \]Is a Homogeneous Linear PDE with constant coefficients.
To solve such equations we process as follows
Step 1: Writing Auxiliary equation
Substitute \[\frac{\partial z}{\partial x}=m,\ \frac{\partial z}{\partial x}=1\]We get auxiliary equation as
\[3m^2+2m-5=0\]Step2 : Finding roots of Auxiliary equation
\[3m^2+2m-5=0\]\[=>3m^2+5m-3m-5=0\]\[=>m\left(3m+5\right)-1\left(3m+5\right)=0\]\[=>\left(3m+5\right)\left(m-1\right)=0\]\[=>m=1,\ -\frac{5}{3}\]These are real and distinct roots.
Step 3 : Writing General solution of the PDE
\[z=c\phi\left(y+1x\right)+d\phi\left(y-\frac{5}{3}x\right)\]Where c and d are arbitrary constat and \[\phi\] is an arbitrary function.