91Ó°ÊÓ

The differential equations are given as:

$\begin{array}{lll}x\text{'}\text{'}+y=1;& x\left(0\right)=1,& x^{\text{'}}\left(0\right)=1\\ x+y^{\text{'}\text{'}}=-1;& y\left(0\right)=1,& y^{\text{'}}\left(0\right)=-1\end{array}$

Given initial value equations are,

$\begin{array}{lll}x\text{'}\text{'}+y=1;& x\left(0\right)=1,& x^{\text{'}}\left(0\right)=\mathrm{1....}\left(\text{1}\right)\\ x+y^{\text{'}\text{'}}=-1;& y\left(0\right)=1,& y^{\text{'}}\left(0\right)=-\mathrm{1....}\left(\text{2}\right)\end{array}$

Taking Laplace transform of equation first we get

$\begin{array}{l}{s}^{2}x\left(s\right)-sx\left(0\right)-x^{\text{'}}\left(0\right)+y\left(s\right)=\frac{1}{s}\\ s^{2}x\left(s\right)-s-1+y\left(s\right)=\frac{1}{s}\\ s^{2}x\left(s\right)+y\left(s\right)=\frac{1+s^2+s}{s}\mathrm{....}\left(3\right)\end{array}$

Taking Laplace transform of equation second we get

$\begin{array}{l}x\left(s\right)+s^{2}y\left(s\right)-sy\left(0\right)-y^{\text{'}}\left(0\right)=\frac{1}{s}\\ x\left(s\right)+s^{2}y\left(s\right)-s+1=-\frac{1}{s}\\ x\left(s\right)+s^{2}y\left(s\right)=\frac{s^2-s-1}{s}\mathrm{.....}\left(4\right)\end{array}$

Solving equation third and fourth we get

$y\left(s\right)=\frac{s^4-s^3-2s^2-s-1}{s(s^4-1)}$

Using partial fraction we can write as

$y\left(s\right)=\frac{s}{s^2+1}-\frac{1}{s-1}+\frac{1}{s}$

Taking inverse Laplace transform we get

$y\left(t\right)=cost-e^t+1$

On solving equation (3) and (4) we, obtain

$x\left(s\right)=\frac{s^4+s^3+s+1}{s(s^4-1)}$

Using partial fraction we can write as

$x\left(s\right)=\frac{s}{s^2+1}+\frac{1}{s-1}-\frac{1}{s}$

Taking inverse Laplace transform we get

$x\left(t\right)=cost+e^t-1$

Hence

$x\left(t\right)=cost+e^t-1,\text{ â¶Ä‰â¶Ä‰}y\left(t\right)=cost-e^t+1$

The final solution is

$x\left(t\right)=cost+e^t-1,\text{ â¶Ä‰â¶Ä‰}y\left(t\right)=cost-e^t+1$