Home page for accesible maths 6 Chapter 6 contents 6.29 Higher order linear differential equations 6.31 Particular integrals

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6.30 Second order linear, constant coefficients

For real constants $a,b,c$ with $a\neq 0$ and a given real function $q(x)$ , the inhomogeneous equation is

a{{d^{2}y}\over{dx^{2}}}+b{{dy}\over{dx}}+cy=q(x)

∗∗

with corresponding homogeneous equation

a{{d^{2}y}\over{dx^{2}}}+b{{dy}\over{dx}}+cy=0.

∗

The complementary function $CF$ is the general solution of $(\ast)$ , as in Thm. 4.45 in MATH101; a particular integral $PI$ is any solution of $(\ast\ast)$ .

Proposition.

The most general solution of $(\ast\ast)$ is given by

y=(CF)+(PI).