## anonymous 5 years ago Can anyone help me to solve this differential equation: (-d/dx)(15*dT(X)/dX)+15*T(x)=528.75

1. anonymous

The LHS is:$-\frac{d}{dx}\left( 15\frac{dT}{dx} \right)+15T=-15\frac{d^2T}{dx^2}+15T$so, letting 528.75=c, and dividing both sides by -15, you have,$T''-T=-\frac{c}{15} \rightarrow T''-\left( T-\frac{c}{15} \right)=0$Let$v=T-\frac{c}{15}$then$v''=T''$and you have$v''-v=0$which is first order, homogeneous, with constant coefficients, so you can assume a solution of the form$v=e^{\lambda x}$Hence$v''=\lambda^2 e^{\lambda x}$and substituting into the d.e., we have$e^{\lambda x}(\lambda^2-1)=0 \rightarrow \lambda = \pm 1$So the solution is$v=c_1e^x+c_2e^{-x}$But $v=T-\frac{c}{15}$so$T=c_1e^x+c_2e^{-x}+\frac{c}{15}$where c=528.75.

2. anonymous

thank you very much...

3. anonymous

You're welcome.

4. anonymous