Editing Separation of variables (section)

=== Example: nonhomogeneous case ===

Suppose the equation is nonhomogeneous,

{{NumBlk|:|<math>\frac{\partial u}{\partial t}-\alpha\frac{\partial^{2}u}{\partial x^{2}}=h(x,t)</math>|{{EqRef|8}}}}

with the boundary condition the same as {{EqNote|2}}.

Expand ''h''(''x,t''), ''u''(''x'',''t'') and ''f''(''x'') into

{{NumBlk|:|<math>h(x,t)=\sum_{n=1}^{\infty}h_{n}(t)\sin\frac{n\pi x}{L},</math>|{{EqRef|9}}}}

{{NumBlk|:|<math>u(x,t)=\sum_{n=1}^{\infty}u_{n}(t)\sin\frac{n\pi x}{L},</math>|{{EqRef|10}}}}

{{NumBlk|:|<math>f(x)=\sum_{n=1}^{\infty}b_{n}\sin\frac{n\pi x}{L},</math>|{{EqRef|11}}}}

where ''h''<sub>''n''</sub>(''t'') and ''b''<sub>''n''</sub> can be calculated by integration, while ''u''<sub>''n''</sub>(''t'') is to be determined.

Substitute {{EqNote|9}} and {{EqNote|10}} back to {{EqNote|8}} and considering the orthogonality of sine functions we get

: <math>u'_{n}(t)+\alpha\frac{n^{2}\pi^{2}}{L^{2}}u_{n}(t)=h_{n}(t),</math>

which are a sequence of [[linear differential equations]] that can be readily solved with, for instance, [[Laplace transform]], or [[Integrating factor]]. Finally, we can get
: <math>u_{n}(t)=e^{-\alpha\frac{n^{2}\pi^{2}}{L^{2}} t} \left (b_{n}+\int_{0}^{t}h_{n}(s)e^{\alpha\frac{n^{2}\pi^{2}}{L^{2}} s} \, ds \right).</math>

If the boundary condition is nonhomogeneous, then the expansion of {{EqNote|9}} and {{EqNote|10}} is no longer valid. One has to find a function ''v'' that satisfies the boundary condition only, and subtract it from ''u''. The function ''u-v'' then satisfies homogeneous boundary condition, and can be solved with the above method.