Home page for accesible maths 6 Chapter 6 contents 6.31 Particular integrals 6.33 Example continued

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6.32 First example

Example.

Find the general solution of the differential equation

\frac{d^{2}y}{dx^{2}}+5\frac{dy}{dx}+6y=6x^{2}+1.

Solution. We first find the CF, i.e. the general solution to the homogeneous equation, using the method from MATH101. The auxiliary equation here is ${s^{2}+5s+6,}$ which has two real roots: ${-2}$ and ${-3.}$ Then the CF is:

y={Ae^{-2x}+Be^{-3x}.}