Solutions to a given nonhomogeneous equation?

**qtpipi** · September 24th 2009, 08:43 PM

let y_p be a particular solution of the nonhomogeneous eqn y'' + py' + qy = f(x) and let y_c be a solution of its associated homogeneous eqn. show that y=y_c + y_p is a solution of the given nonhomogeneous eqn.

**chisigma** · September 25th 2009, 02:01 AM

On the basis of your hypotheses is...

$y_{c}^{''} + p \cdot y_{c}^{'} + q\cdot y_{c} =0$

$y_{p}^{''} + p \cdot y_{p}^{'} + q\cdot y_{p} = f(x)$

... so that setting $y(x)= y_{c} (x) + y_{p} (x)$ , because the derivative is a linear operator, you have...

$y^{''} + p \cdot y^{'} + q\cdot y = f(x)$

Kind regards

$\chi$ $\sigma$

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