Solve the integral of differential equation

[izen]

Homework Statement

Solve the integral of differential equation

∫$^{t}_{0}$ y($\tau$) d$\tau$ -$\acute{y}$ (t) = t

for t≥0 with y(0)=4

The Attempt at a Solution

take laplace both sides

$\frac{Y}{s}$ - sY + 4 = $\frac{1}{s^{2}}$

Y $\frac{1 - s^{2}}{s}$ =$\frac{1}{s^{2}}$ - 4

Y $\frac{s^{2}-1}{s}$ =- $\frac{1}{s^{2}}$ + 4

Y = - $\frac{1}{s(s^{2}-1)}$ + $\frac{4s}{s^{2}-1}$

Partial fraction - $\frac{1}{s(s^{2}-1)}$

$\frac{1}{s}$ -$\frac{s}{s^{2}-1}$ +$\frac{4s}{s^{2}-1}$

inverse laplace transform

1-cos(t)+4cos(t) = > 1+3cos(t)

but the answer is 1+$\frac{3}{2}$ (e$^{t}$ + e$^{-t}$)

Please check my solution

Thank you

 

[haruspex]

$\frac{1}{s}$ -$\frac{s}{s^{2}-1}$ +$\frac{4s}{s^{2}-1}$

inverse laplace transform

1-cos(t)+4cos(t) = > 1+3cos(t)

I think you mean cosh, not cos. It would be cos if the denominator were s²+1.

 

[HallsofIvy]

I have never like "Laplace Transform"- too mechanical. In particular, for this problem, using the Laplace Transform is using a cannon to kill a fly.

Here is how I would do this problem: differentiate both sides of the equation, with respect to t, to get rid of the integral: y(t)- y''(t)= 1 or y''- y= -1. We have initial conditions y(0)= 0 and y'(0)= 0 (from setting t=0 in the orginal equation). That's a very simple linear equation with constant coefficients. The characteristic equation has real roots so the solution will involve exponentials (and so cosh as haruspex says) not trig functions.

 

[izen]

ohh thanks 1+3cosh(t) and then i can use the complex exponential formula to change cosh(t) to 1/2 (e^t + e^-t) thanks haruspex :)

HallsofIvy << I like your saying "using the Laplace Transform is using a cannon to kill a fly" could not agree more thanks for the the comment :)

 

[izen]

I have never like "Laplace Transform"- too mechanical. In particular, for this problem, using the Laplace Transform is using a cannon to kill a fly.
.

My teacher convinces the class that the Laplace transform is so powerful but like your saying maybe it is too powerful on such this simple problem. :)