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Question Number 1056 by 123456 last updated on 25/May/15

$$y^{″}+y=8cos\left(t\right)$$$$y\left(0\right)=+1$$$$y^{\prime }\left(0\right)=-1$$

Commented by prakash jain last updated on 26/May/15

$$\mathrm{characterstic}\mathrm{equation}$$$$r^2+1=0$$$$y_h=c_1\cos t+c_2\sin t$$

Answered by prakash jain last updated on 26/May/15

$$y_p=kt\sin t$$$$y_p^{\prime }=kt\cos t+k\sin t$$$$y_p^{″}=-kt\sin t+k\cos t+k\cos t$$$$y_p^{″}+y=2k\cos t=8\cos t\Rightarrow k=4$$$$y=y_p+y_h=c_1\cos t+c_2\sin t+4t\cos t$$$$y\left(0\right)=1\Rightarrow c_1=1$$$$y^{\prime }=4t\cos t+4\sin t-c_1\sin t+c_2\cos t$$$$y^{\prime }\left(0\right)=-1\Rightarrow c_2=-1$$$$y=\cos t-\sin t+4t\sin t$$

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