Home page for accesible maths 6 Chapter 6 contents 6.36 Example: exceptional case 6.38 Summary of method

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6.37 Several examples

Example.

Which form should the particular integral take for each of the following second order equations?

a) $y^{\prime\prime}+4y=x^{2};\;\;\hbox{b)}\;y^{\prime\prime}+4y=e^{2x};\;\;\hbox{% c)}\;y^{\prime\prime}+4y=\cos 2x$ ;

d) $y^{\prime\prime}+5y^{\prime}+6y=e^{-2x};\;\;\hbox{e)}\;y^{\prime\prime}+2y^{% \prime}+2y=e^{-x}\cos x;\;\;\hbox{f)}\;y^{\prime\prime}-2y^{\prime}=x$ .

For cases (a)-(c) the CF is $A\cos 2x+B\sin 2x$ , so only (c) is an exceptional case. Thus for (a), the PI has the form $b_{2}x^{2}+b_{1}x+b_{0}$ ; for (b) it is of the form $Ce^{2x}$ , for (c) it is of the form $x(H\cos 2x+K\sin 2x)$ .

In (d), the auxiliary equation is $s^{2}+5s+6={(s+2)(s+3),}$ so this is also an exceptional case, hence the PI has the form $Cxe^{-2x}$ .

For (e), the CF is $(A\cos x+B\sin x)e^{-x}$ so this is also an exceptional case, hence the PI has the form $(H\cos x+K\sin x)xe^{-x}$ .

Finally, (f) is also an exceptional case as $1$ is a solution to the homogeneous equation, so the PI is of the form $Cx^{2}+Dx$ .