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]

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

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