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# Solve simultaneous equation dx/dt=3x+5y , dy/dt=-x-y , x=5 , y=-3...

Solve simultaneous equation dx/dt=3x+5y , dy/dt=-x-y , x=5 , y=-3

$\frac{dx}{dt}=3x+5y\phantom{\rule{0ex}{0ex}}\frac{dy}{dt}=-x-y\phantom{\rule{0ex}{0ex}}$

$⇒\mathbit{y}\left(t\right)\mathbf{=}\left({C}_{1}sin\left(t\right)+{C}_{2}cos\left(t\right)\right){\mathbit{e}}^{\mathbf{t}}$

$⇒y\left(t\right)=\left({C}_{1}\mathrm{sin}\left(t\right)+{C}_{2}\mathrm{cos}\left(t\right)\right){e}^{t}$

$⇒\frac{dy}{dt}=\frac{d}{dt}\left[\left({C}_{1}\mathrm{sin}\left(t\right)+{C}_{2}\mathrm{cos}\left(t\right)\right){e}^{t}\right]\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dt}={e}^{t}\frac{d}{dt}\left[\left({C}_{1}\mathrm{sin}\left(t\right)+{C}_{2}\mathrm{cos}\left(t\right)\right)\right]+\left({C}_{1}\mathrm{sin}\left(t\right)+{C}_{2}\mathrm{cos}\left(t\right)\right)\frac{d}{dt}\left[{e}^{t}\right]\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dt}=\left({C}_{1}\mathrm{cos}\left(t\right)-{C}_{2}\mathrm{sin}\left(t\right)\right){e}^{t}+\left({C}_{1}\mathrm{sin}\left(t\right)+{C}_{2}\mathrm{cos}\left(t\right)\right){e}^{t}\phantom{\rule{0ex}{0ex}}⇒\frac{dy}{dt}=\left({C}_{1}\left(\mathrm{cos}\left(t\right)+\mathrm{sin}\left(t\right)\right)+{C}_{2}\left(\mathrm{cos}\left(t\right)-\mathrm{sin}\left(t\right)\right)\right){e}^{t}\phantom{\rule{0ex}{0ex}}$

$⇒x=-\left({C}_{1}\left(\mathrm{cos}\left(t\right)+\mathrm{sin}\left(t\right)\right)+{C}_{2}\left(\mathrm{cos}\left(t\right)-\mathrm{sin}\left(t\right)\right)\right){e}^{t}-\left({C}_{1}\mathrm{sin}\left(t\right)+{C}_{2}\mathrm{cos}\left(t\right)\right){e}^{t}\phantom{\rule{0ex}{0ex}}⇒\mathbit{x}\left(t\right)\mathbf{=}\mathbf{-}\left({C}_{1}\left(cos\left(t\right)+2sin\left(t\right)\right)+{C}_{2}\left(2cos\left(t\right)-sin\left(t\right)\right)\right){\mathbit{e}}^{\mathbf{t}}\phantom{\rule{0ex}{0ex}}$

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