# First order ODE IVP use in Heat equation of ball with heat source.

1. Jun 29, 2010

### yungman

This should be a simple ODE problem inside Heat problem of ball with steady state heat source. I am going to present the question first and then I am going to put in a little bit of the formulas to bring this into the big picture:

$$B'_{jnm}(t) + \lambda^2_{nj} c^2 B_{jnm}(t) = q_{jnm}$$ (1)

and

$$B_{jnm}(0)= f_{jnm}$$(2)

I want to solve for $B_{jnm}(t)$.

(1) and (2) form a simple first order ODE with initial value problem. I cannot get the result from the book. This is my derivations:

Using variation of parameter: Let $B_{jnm}(t)= B_{jnmc} + B_{jnmp}$

$B_{jnmc}$ is solution of associate homogeneous equation and $B_{jnmp}$ is the particular solution of the non-homogeneous DE.

$$B'_{jnmc} + \lambda^2_{nj} c^2 B_{jnmc} = 0 \Rightarrow\; B_{jnmc} = C_1 e^{-\lambda^2_{nj} c^2 t}$$

$$B_{jnmp} = u B_{jnmc} = u C_1 e^{-\lambda^2_{nj} c^2 t}$$

$$B_{jnmp} = C_1 e^{-\lambda^2_{nj} c^2 t} \int \frac{g_{jnm}}{C_1} e^{\lambda^2_{nj} c^2 t} dt = C_1 [\frac{g_{jnm}}{\lambda^2_{nj} c^2}]$$

$$B_{jnm}(t) = B_{jnmc} + B_{jnmp} = C_1 e^{-\lambda^2_{nj} c^2 t} + \frac{g_{jnm}} {\lambda^2_{nj} c^2}$$

Initial condition:

$$B_{jnm}(0) = f_{jnm} = C_1 + \frac{g_{jnm}} {\lambda^2_{nj} c^2}$$

$$\Rightarrow C_1 = f_{jnm} - \frac{q_{jnm}} {\lambda^2_{nj} c^2}$$

Last edited: Jun 30, 2010
2. Jun 29, 2010

### yungman

I found out what I did wrong already. It was so stupid!! I forgot the constant. I'll edit the original post.