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Question Number 107108 by Ar Brandon last updated on 08/Aug/20

$${\mathrm{y}}^{″}+\mathrm{y}=\frac{1}{\mathrm{cosx}}$$

Commented by mohammad17 last updated on 08/Aug/20

$$(m^2-i^2)=0\Rightarrow m=\mp i$$$$Yc=c_{1}cosx+c_{2}sinx$$$$let:Yp=u_1{v}_1+u_2{v}_2$$$$u_1=cosx,u_2=sinx\Rightarrow u_1^{\prime }=-sinx,u_2^{\prime }=cosx$$$$D=u_1{u}_2^{\prime }-u_2{u}_1^{\prime }={cos}^{2}x+{sin}^{2}x=1$$$$$$$$v_1=\int \frac{-sinx}{cosx}dx=ln\mid cosx\mid$$$$v_2=\int \frac{cosx}{cosx}dx=x$$$$\therefore Yp=cosxln\mid cosx\mid +xsinx$$$$$$$$Y=Yc+Yp=c_{1}cosx+c_{2}sinx+cosxln\mid cosx\mid +xsinx$$$$$$$$M.S.MohammadTaha$$

Commented by Ar Brandon last updated on 08/Aug/20

Thanks��

Commented by mohammad17 last updated on 08/Aug/20

$$welcome$$

Answered by mathmax by abdo last updated on 08/Aug/20

$${\mathrm{y}}^{{}^{″}}+\mathrm{y}=\frac{1}{\mathrm{cosx}}$$$$\mathrm{h}\to {\mathrm{y}}^{{}^{″}}+\mathrm{y}=0\Rightarrow {\mathrm{r}}^2+1=0\Rightarrow \mathrm{r}=\stackrel{-}{+}1\Rightarrow {\mathrm{y}}_{\mathrm{h}}=\mathrm{acosx}+\mathrm{bsinx}={\mathrm{au}}_1+{\mathrm{bu}}_2$$$$\mathrm{W}({\mathrm{u}}_1,{\mathrm{u}}_2)=\left|\begin{array}{c}\mathrm{cosx}\mathrm{sinx}\\ -\mathrm{sinx}\mathrm{cosx}\end{array}\right|={\cos }^2\mathrm{x}+{\sin }^2\mathrm{x}=1$$$${\mathrm{W}}_1=\left|\begin{array}{c}0\mathrm{sinx}\\ \frac{1}{\mathrm{cosx}}\mathrm{cosx}\end{array}\right|=-\frac{\mathrm{sinx}}{\mathrm{cosx}}$$$${\mathrm{W}}_2=\left|\begin{array}{c}\mathrm{cosx}0\\ -\mathrm{sinx}\frac{1}{\mathrm{cosx}}\end{array}\right|=1$$$${\mathrm{v}}_1=\int \frac{{\mathrm{w}}_1}{\mathrm{w}}\mathrm{dx}=-\int \frac{\mathrm{sinx}}{\mathrm{cosx}}\mathrm{dx}=\ln \mid \mathrm{cosx}\mid$$$${\mathrm{v}}_2=\int \frac{{\mathrm{w}}_2}{\mathrm{w}}\mathrm{dx}=\int \mathrm{dx}=\mathrm{x}\Rightarrow {\mathrm{y}}_{\mathrm{p}}={\mathrm{u}}_1{\mathrm{v}}_1+{\mathrm{u}}_2{\mathrm{v}}_2=\mathrm{cosxln}\mid \mathrm{cosx}\mid +\mathrm{xsinx}$$$$\mathrm{the}\mathrm{general}\mathrm{solution}\mathrm{is}\mathrm{y}={\mathrm{y}}_{\mathrm{h}}+{\mathrm{y}}_{\mathrm{p}}$$$$\Rightarrow \mathrm{y}=\mathrm{cosxln}\mid \mathrm{cosx}\mid +\mathrm{xsinx}+\mathrm{acosx}+\mathrm{bsinx}$$

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