Question

Reduce into canonical form
4Uxx+5Uxy+Uyy+Ux+Uy=0

Answer 1

$4Uxx+5Uxy+Uyy+Ux+Uy=0\\ or, 4r+5s+t+p+q=0\\ where R=4,S=5,T=1.\\ \text{therefore, characteristic equation is }R\lambda^2+S\lambda+T=0\\ 4\lambda^2+5\lambda+1=0\\ \lambda=-1/4,-1\\ Then,\\ \text{the characteristics curve of given pde is the solution of these equation}\\ \frac{dy}{dx}+\lambda_1=0 --------(1)\\ and\\ \frac{dy}{dx}-\lambda_2=0---------(2) \\ \text{from (1),we get}\\ c_1=y-(x/4) (=m)\\ \text{from (2),we get}\\ c_1=y-x (=n)\\ \text{Now, find p,q,r,s,t and substitute in the given pde.}\\ p=Ux=\frac{-1}{4}Um-Un\\ q=Uy=Um+Un\\ r=Uxx=\frac{1}{16}Umm+\frac{1}{2}Umn+Unn \\ s=\frac{-1}{4}Umm-\frac{5}{4}Umn-Unn\\ t=Umm+2Umn+Unn\\ \text{substitute in the given pde, we get}\\ Umm=\frac{Um}{3}\\ \text{This is required canonical form.}$