Answer to Question #294429 in Differential Equations for Lucky

Question #294429

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


1
Expert's answer
2022-02-08T15:27:29-0500

we shall solve the partial differential equation

(3D2+10DD+3D2)z=exy(3D^{2}+10DD^{\prime}+3D^{\prime^{2}})z=e^{x-y}

The auxilliary equation is

3m2+10m+3=0(3m+1)(m+3)=03m^{2}+10m+3=0\\(3m+1)(m+3)=0

3m+1=03m+1=0 or m+3=0m+3=0

m=13,3m=-\frac{1}{3},3


C.F=f1(y13x)+f2(y3x)C.F=f_{1}(y-\frac{1}{3}x)+f_{2}(y-3x)


We have the P.I to be

P.I=13D2+10DD+3D2exyP.I=\frac{1}{3D^{2}+10DD^{\prime}+3D^{\prime^{2}}}e^{x-y}


=13(1)2+10(1)(1)+3(1)2exy=\frac{1}{3(1)^{2}+10(1)(-1)+3(-1)^{2}}e^{x-y}


=1310+3exy=\frac{1}{3-10+3}e^{x-y}


=14exy=-\frac{1}{4}e^{x-y}

z=f1(y13x)+f2(y3x)14exyz=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.


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment