91Ó°ÊÓ

The given expressions are $y^{\text{'}\text{'}}−a^{2}y=e^{−t}$.

Integration by partsor partial integration is a process that finds theintegralof aproductoffunctionsin terms of the integral of the product of theirderivativeandanti-derivative.

Use the function.

$$\begin{gathered} \begin{array}{c}y=\left\{\begin{array}{l}0\text{ â¶Ä„â¶Ä„â¶ÄŠ}0<t^{\text{'}}<t\\ {∫}_0^{\mathrm{∞}}\frac{1}{a}\sin a\left(t−t^{\text{'}}\right)\text{ â¶Ä„â¶Ä„â¶ÄŠ}0<t^{\text{'}}<t\end{array}\right.\\ y\left(t\right)={∫}_0^{t}d{t}^{\text{'}}\frac{1}{\mathrm{Î\pm }}\sin a\left(t−t^{\text{'}}\right)e^{−f^{\text{'}}}\end{array} \\ \mathrm{Thus},\mathrm{we}\mathrm{have} \\ \begin{array}{l}\mathrm{I}={∫}_0^{\mathrm{t}}\frac{1}{\mathrm{a}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\\ =\left({\left.\mathrm{uv}\right|}_{\mathrm{t}}^0−{∫}_0^{\mathrm{t}}\mathrm{vdu}\right.\\ =\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−{∫}_0^{\mathrm{t}}\mathrm{acosh}\left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\right.\\ \mathrm{Again},\mathrm{we}\mathrm{use}\mathrm{integration}\mathrm{by}\mathrm{parts}\mathrm{to}\mathrm{evaluate}\mathrm{the}\mathrm{last}\mathrm{integral},\mathrm{hence}\mathrm{we}\mathrm{have}\\ \begin{array}{l}{\mathrm{I}}_1={∫}_0^{\mathrm{t}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\\ =\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0\right.−{∫}_0^{\mathrm{t}}\mathrm{asinh}\left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\end{array}\\ \mathrm{Thus},\mathrm{we}\mathrm{have}\\ \begin{array}{l}\mathrm{I}=\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−\mathrm{a}{∫}_0^{\mathrm{t}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}=\right.\\ \mathrm{I}=\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−\mathrm{a}\left(\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−\mathrm{aI}\right)\right.\right.\end{array}\\ \begin{array}{l}\mathrm{I}=(−0+1â‹\dots \sinh (\mathrm{at}\left)\right)−\mathrm{a}\left[\left(−{\mathrm{e}}^{\mathrm{t}}â‹\dots 1+\cosh \left(\mathrm{at}\right)\right)−\mathrm{aI}\right]\\ \mathrm{I}=\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)+{\mathrm{a}}^2\mathrm{I}\end{array}\\ \mathrm{Hence}\mathrm{the}\mathrm{evaluation}\mathrm{of}\mathrm{the}\mathrm{integral}\mathrm{I}\mathrm{is}\mathrm{thus},\\ \begin{array}{l}\mathrm{I}\left(1−{\mathrm{a}}^2\right)=\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)\\ \mathrm{I}=\frac{\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)}{1−{\mathrm{a}}^2}\end{array}\\ \mathrm{And},\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{differential}\mathrm{equation}\mathrm{is}\mathrm{given}\mathrm{by}\\ \mathrm{y}\left(\mathrm{t}\right)=\frac{1}{\mathrm{a}}\mathrm{I}\\ \mathrm{Thus},\mathrm{we}\mathrm{have}\\ \mathrm{y}\left(\mathrm{t}\right)=\frac{\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)}{\mathrm{a}\left(1−{\mathrm{a}}^2\right)}\\ \mathrm{y}\left(\mathrm{t}\right)=\frac{\mathrm{acosh}\left(\mathrm{at}\right)−{\mathrm{ae}}^{\mathrm{t}}−\sinh \left(\mathrm{at}\right)}{\mathrm{a}\left({\mathrm{a}}^2−1\right)}\\ \mathrm{This},\mathrm{is}\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{differential}\mathrm{equation}\mathrm{for}\mathrm{all}\mathrm{t}>{\mathrm{t}}^{\text{'}}>0\mathrm{else},\mathrm{the}\mathrm{solution}\mathrm{is}\mathrm{zero}.\end{array} \end{gathered}$$uncaught exception: Invalid chunk

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$$\begin{gathered} \mathrm{Thus},\mathrm{we}\mathrm{have} \\ \begin{array}{l}\mathrm{I}={∫}_0^{\mathrm{t}}\frac{1}{\mathrm{a}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\\ =\left({\left.\mathrm{uv}\right|}_{\mathrm{t}}^0−{∫}_0^{\mathrm{t}}\mathrm{vdu}\right.\\ =\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−{∫}_0^{\mathrm{t}}\mathrm{acosh}\left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\right.\\ \mathrm{Again},\mathrm{we}\mathrm{use}\mathrm{integration}\mathrm{by}\mathrm{parts}\mathrm{to}\mathrm{evaluate}\mathrm{the}\mathrm{last}\mathrm{integral},\mathrm{hence}\mathrm{we}\mathrm{have}\\ \begin{array}{l}{\mathrm{I}}_1={∫}_0^{\mathrm{t}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\\ =\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0\right.−{∫}_0^{\mathrm{t}}\mathrm{asinh}\left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}\end{array}\end{array} \end{gathered}$$

$\begin{array}{l}\mathrm{Thus},\mathrm{we}\mathrm{have}\\ \begin{array}{l}\mathrm{I}=\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−\mathrm{a}{∫}_0^{\mathrm{t}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right){\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}{\mathrm{dt}}^{\text{'}}=\right.\\ \mathrm{I}=\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\sinh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−\mathrm{a}\left(\left(−{\left.{\mathrm{e}}^{−{\mathrm{t}}^{\text{'}}}\cosh \left(\mathrm{a}\left(\mathrm{t}−{\mathrm{t}}^{\text{'}}\right)\right)\right|}_{\mathrm{t}}^0−\mathrm{aI}\right)\right.\right.\end{array}\\ \begin{array}{l}\mathrm{I}=(−0+1â‹\dots \sinh (\mathrm{at}\left)\right)−\mathrm{a}\left[\left(−{\mathrm{e}}^{\mathrm{t}}â‹\dots 1+\cosh \left(\mathrm{at}\right)\right)−\mathrm{aI}\right]\\ \mathrm{I}=\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)+{\mathrm{a}}^2\mathrm{I}\end{array}\\ \mathrm{Hence}\mathrm{the}\mathrm{evaluation}\mathrm{of}\mathrm{the}\mathrm{integral}\mathrm{I}\mathrm{is}\mathrm{thus},\\ \begin{array}{l}\mathrm{I}\left(1−{\mathrm{a}}^2\right)=\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)\\ \mathrm{I}=\frac{\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)}{1−{\mathrm{a}}^2}\end{array}\\ \mathrm{And},\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{differential}\mathrm{equation}\mathrm{is}\mathrm{given}\mathrm{by}\\ \mathrm{y}\left(\mathrm{t}\right)=\frac{1}{\mathrm{a}}\mathrm{I}\\ \mathrm{Thus},\mathrm{we}\mathrm{have}\\ \mathrm{y}\left(\mathrm{t}\right)=\frac{\sinh \left(\mathrm{at}\right)+{\mathrm{ae}}^{\mathrm{t}}−\mathrm{acosh}\left(\mathrm{at}\right)}{\mathrm{a}\left(1−{\mathrm{a}}^2\right)}\\ \mathrm{y}\left(\mathrm{t}\right)=\frac{\mathrm{acosh}\left(\mathrm{at}\right)−{\mathrm{ae}}^{\mathrm{t}}−\sinh \left(\mathrm{at}\right)}{\mathrm{a}\left({\mathrm{a}}^2−1\right)}\\ \mathrm{This},\mathrm{is}\mathrm{the}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{given}\mathrm{differential}\mathrm{equation}\mathrm{for}\mathrm{all}\mathrm{t}>{\mathrm{t}}^{\text{'}}>0\mathrm{else},\mathrm{the}\mathrm{solution}\mathrm{is}\mathrm{zero}.\end{array}$