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:(adsbygoogle = window.adsbygoogle || []).push({});

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

and

[tex] B_{jnm}(0)= f_{jnm}[/tex](2)

I want to solve for [itex]B_{jnm}(t)[/itex].

(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 [itex] B_{jnm}(t)= B_{jnmc} + B_{jnmp} [/itex]

[itex]B_{jnmc} [/itex] is solution of associate homogeneous equation and [itex]B_{jnmp} [/itex] is the particular solution of the non-homogeneous DE.

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

[tex] B_{jnmp} = u B_{jnmc} = u C_1 e^{-\lambda^2_{nj} c^2 t} [/tex]

[tex] 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}] [/tex]

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

Initial condition:

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

[tex] \Rightarrow C_1 = f_{jnm} - \frac{q_{jnm}} {\lambda^2_{nj} c^2} [/tex]

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**