Question: Use equation (12.6) to solve Problem 10.18.

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.

$y\left(t\right)={\int }_0^t\frac{1}{\omega }\cdot \sin \omega \left(t-t^{\text{'}}\right)f\left(t^{\text{'}}\right)d{t}^{\text{'}}$

$\begin{array}{c}y\left(t\right)=\left\{\begin{array}{ll}{\int }_0^t\frac{1}{\omega }\cdot \sin \omega \left(t-t^{\text{'}}\right)\times 1d{t}^{\text{'}}& 0<t^{\text{'}}\\ {\int }_0^t\frac{1}{\omega }\cdot \sin \omega \left(t-t^{\text{'}}\right)\times 0d{t}^{\text{'}}& t^{\text{'}}>a\end{array}\right.\\ y\left(t\right)=\left\{\begin{array}{c}{\int }_0^t\frac{1}{\omega }\cdot \sin \omega \left(t-t^{\text{'}}\right)d{t}^{\text{'}}0<t^{\text{'}}<a\\ 0t^{\text{'}}>a\end{array}\right.\\ y\left(t\right)=\left\{\begin{array}{c}\frac{1}{{\omega }^2}\cdot \cos \omega \left(t-t^{\text{'}}\right)d{t}^{\text{'}}0<t^{\text{'}}<a\\ 0t^{\text{'}}>a\end{array}\right.\end{array}$

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$y\left(t\right)=\left\{\begin{array}{cc}\frac{1}{{\omega }^2}\cdot \cos \omega \left(t-t^{\text{'}}\right)d{t}^{\text{'}}& 0<t^{\text{'}}<a\\ 0t^{\text{'}}>a& \end{array}\right.$