91Ó°ÊÓ

The given expressions are$f\left(t\right)=\left\{\begin{array}{ll}0;& 0<t<a\\ 1;& y_0=y_0^{\text{'}}=0\end{array}.\right.$

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

Use the function.

$f\left(t\right)=\left\{\begin{array}{ll}0;& 0<t<a\\ 1;& y_0=y_0^{\text{'}}=0\end{array}\right.$

Use the integration.

for the function f(t) using the Heaviside function, thus a representation given would be

$f\left(t\right)=u\left(t\right)−u(t−a)$

It must be noted that it is valid to use equation (12.6) with such differential equation, as it is given that the boundaries values for this differential equation is

$y_0^{\text{'}}=y_0=0$

And, hence we start using equation to find the solution of this differential equation, thus

Use the Euler equation.

$y\left(t\right)={∫}_0^t\frac{1}{\mathrm{Ó\neg }}â‹\dots \sin \mathrm{Ó\neg }\left(t−t^{\text{'}}\right)f\left(t^{\text{'}}\right)d{t}^{\text{'}}$

$\begin{array}{c}y\left(t\right)=\left\{\begin{array}{ll}{∫}_0^t\frac{1}{\mathrm{Ó\neg }}â‹\dots \sin \mathrm{Ó\neg }\left(t−t^{\text{'}}\right)×1d{t}^{\text{'}}& 0<t^{\text{'}}\\ {∫}_0^t\frac{1}{\mathrm{Ó\neg }}â‹\dots \sin \mathrm{Ó\neg }\left(t−t^{\text{'}}\right)×0d{t}^{\text{'}}& t^{\text{'}}>a\end{array}\right.\\ y\left(t\right)=\left\{\begin{array}{c}{∫}_0^t\frac{1}{\mathrm{Ó\neg }}â‹\dots \sin \mathrm{Ó\neg }\left(t−t^{\text{'}}\right)d{t}^{\text{'}}\text{ â¶Ä„â¶Ä„â¶ÄŠ}0<t^{\text{'}}<a\\ 0\text{ â¶Ä„â¶Ä„â¶ÄŠ}{t}^{\text{'}}>a\end{array}\right.\\ y\left(t\right)=\left\{\begin{array}{c}\frac{1}{{\mathrm{Ó\neg }}^2}â‹\dots \cos \mathrm{Ó\neg }\left(t−t^{\text{'}}\right)d{t}^{\text{'}}\text{ â¶Ä„â¶Ä„â¶ÄŠ}0<t^{\text{'}}<a\\ 0\text{ â¶Ä„â¶Ä„â¶ÄŠ}{t}^{\text{'}}>a\end{array}\right.\end{array}$

Thus, the value of$f\left(t\right)=\left\{\begin{array}{ll}0;& 0<t<a\\ 1;& y_0=y_0^{\text{'}}=0\end{array}\right.$ value of is$y\left(t\right)=\left\{\begin{array}{cc}\frac{1}{{\mathrm{Ó\neg }}^2}â‹\dots \cos \mathrm{Ó\neg }\left(t−t^{\text{'}}\right)d{t}^{\text{'}}& 0<t^{\text{'}}<a\\ 0\text{ â¶Ä„â¶Ä„â¶ÄŠ}{t}^{\text{'}}>a& \end{array}\right.$