# Making homogenous a wave equation

#### Markov

##### Member
Solve

\begin{aligned} & {{u}_{tt}}={{u}_{xx}}+t,\text{ }t>0,\text{ }x\in \mathbb R, \\ & u(x,0)=x \\ & {{u}_{t}}(x,0)=1. \end{aligned}

Okay first I should set $v(x,t)=u(x,t)-\dfrac16 t^3,$ then $u(x,t)=v(x,t)+\dfrac16 t^3$ so $u_{tt}=v_{tt}+t$ and $u_{xx}=v_{xx}$ so $v_{tt}+t=v_{xx}+t\implies v_{tt}=v_{xx},$ and $u(x,0)=v(x,0)$ and $u_t(x,0)=v_t(x,0),$ so I need to solve

\begin{aligned} & {{v}_{tt}}={{v}_{xx}},\text{ }t>0,\text{ }x\in \mathbb R, \\ & v(x,0)=x \\ & {{v}_{t}}(x,0)=1. \end{aligned}

which is a simple application of the formula and then once found $v$ the problem is solved! Is it correct?

#### Jester

##### Well-known member
MHB Math Helper
So far so good.

Last edited by a moderator: