Q10E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x′, y′, etc., denotes differentiation with respect to t; so does the symbol D.

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

Step-by-Step Solution

Answer

The solution is 

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

Step 1: Given information

The differential equations are given as:

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

Step 2: Apply the Laplace transform

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 

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

Step 3: Solve the third and fourth equation

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 

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

Step 4: conclusion

The final solution is 

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