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Question Number 27028 by sorour87 last updated on 01/Jan/18

$$y^{\left(2\right)}+4y=\sinh x\times \sin 2x$$

Answered by prakash jain last updated on 02/Jan/18

$${\lambda }^2+4=0$$$$\lambda =\pm 2i$$$$y_h=c_1\cos 2x+c_2\sin 2x$$$$y_1=\cos 2x,y_2=\sin 2x$$$$\mathcal{W}=\left|\begin{array}{cc}\cos 2x& \sin 2x\\ -2\sin 2x& 2\cos 2x\end{array}\right|=2$$$$v_1=-\int \frac{\sinh x\times \sin 2x}{2}\times \sin 2xdx$$$$v_1=-\frac{1}{2}\int \frac{1}{2}(1-\cos 4x)\times \sinh xdx$$$$=-\frac{1}{4}\int \sinh x-\int \cos 4x\sinh xdx$$$$v_2=\int \frac{\sinh x\times \sin 2x}{2}\times \cos 2xdx$$$$y_p=v_1{y}_1+v_2{y}_2$$$$y=y_h+y_p$$

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