- #1
Math100
- 819
- 231
- Homework Statement
- Find the general solution of the partial differential equation ## u_{xx}+6u_{x}+9u=0 ##.
- Relevant Equations
- None.
Find the general solution of the partial differential equation ## u_{xx}+6u_{x}+9u=0 ##.
Here's my work:
Let ## u(x, y)=e^{rx} ##.
Then ## u_{x}(x, y)=re^{rx} ## and we get that ## u_{xx}(x, y)=r^2e^{rx} ##.
By using direct substitution of ## u=e^{rx}, u_{x}=re^{rx} ## and ## u_{xx}=r^2e^{rx} ##,
we have that ## u_{xx}+6u_{x}+9u=0\implies r^2e^{rx}+6re^{rx}+9e^{rx}=0\implies
e^{rx}(r^2+6r+9)=0\implies r^2+6r+9=0\implies (r+3)^2=0 ##.
Thus, ## r=-3 ##.
Therefore, the general solution of the partial differential equation ## u_{xx}+6u_{x}+9u=0 ## is
## u(x, y)=A(y)e^{-3x}+B(y)\cdot xe^{-3x} ## where ## A, B ## are arbitrary functions of ## y ##.
Above is my work with the answer. May someone please check/verify to see if everything is correct? Also, I want to know where does the substitution of ## u(x, y)=e^{rx} ## comes from, although I was just told to use it without any definition, lemma, theorem, etc. As for the characteristic equations with repeated roots like the given partial differential equation above, will its general solution always of the form ## u(x, y)=A(y)e^{rx}+B(y)\cdot xe^{rx} ## where ## A, B ## are arbitrary functions of ## y ##?
Here's my work:
Let ## u(x, y)=e^{rx} ##.
Then ## u_{x}(x, y)=re^{rx} ## and we get that ## u_{xx}(x, y)=r^2e^{rx} ##.
By using direct substitution of ## u=e^{rx}, u_{x}=re^{rx} ## and ## u_{xx}=r^2e^{rx} ##,
we have that ## u_{xx}+6u_{x}+9u=0\implies r^2e^{rx}+6re^{rx}+9e^{rx}=0\implies
e^{rx}(r^2+6r+9)=0\implies r^2+6r+9=0\implies (r+3)^2=0 ##.
Thus, ## r=-3 ##.
Therefore, the general solution of the partial differential equation ## u_{xx}+6u_{x}+9u=0 ## is
## u(x, y)=A(y)e^{-3x}+B(y)\cdot xe^{-3x} ## where ## A, B ## are arbitrary functions of ## y ##.
Above is my work with the answer. May someone please check/verify to see if everything is correct? Also, I want to know where does the substitution of ## u(x, y)=e^{rx} ## comes from, although I was just told to use it without any definition, lemma, theorem, etc. As for the characteristic equations with repeated roots like the given partial differential equation above, will its general solution always of the form ## u(x, y)=A(y)e^{rx}+B(y)\cdot xe^{rx} ## where ## A, B ## are arbitrary functions of ## y ##?