After a semester full of discovering new ways to solve differential equations, we have now come to the end. In block 6 we will cover the last method this semester of solving differential equations. We will be taking a look at the laplase transform to linear diff. eq. This method uses a transformation to change the domain of the differential equations. To transform and then convert the problems back and forth a list of identities can be used. Another way is to simply solve the problems using matlab or mathematica.
In this block I will be choosing 6 examples in which will be completed. 2 first-order equations, 2 second-order equations, and 2 linear systems will be chosen in order to show the use and function of these laplase transforms
laplasesolver.m:
function [x] = laplasesolver (f,y_final,y_mult)
syms s t
laplace(f,t,s);
b=1/(s+(y_mult))
x=ans+(y_final);
figure
ezplot(x,[-20,20])
final=x*(b)
ilaplace(final,s,t);
a=ans;
figure
ezplot(a,[10,15])
end
FIRST ORDER EQUATION EXAMPLES
1. $\latex \frac{dy}{dt}+y=3cos(t)$ ,y(0)=-1
>> laplasesolver(3*cos(t),-1,1)
b =
1/(s + 1)
final =
((3*s)/(s^2 + 1) – 1)/(s + 1)
ans =
(3*cos(t))/2 – 5/(2*exp(t)) + (3*sin(t))/2
Using ilaplace to transform the function to the time domanin
$\latex y(t)=7cos(4*t)$
2. $\latex\frac{dy}{dx}-y=e^(3t), y(0)=2 $
entering this into the m file produces:
b =
1/(s – 1)
final =
(1/(s – 3) + 2)/(s – 1)
ans =
exp(3*t)/2 + (3*exp(t))/2
solution graph:
SECOND ORDER EQUATION EXAMPLES:
1. $\latex \frac{d^2y}{dt^2}+16y=0$ , y(0)=7, y’(0)=0
$\latex L{\frac{d^2y}{dt^2}}=s^2L{y(s)}-7s$
$\latex s^2L{y(s)}-7s+16L{y(s)}=0$
$\latex (s^2+4)L{y(s)}=7s$
$\latex L{y(s)}=\frac{7s}{(s^2+16)}$
Using ilaplace to transform the function yeilds the solution:
$\latex y(t)=7cos(4*t)$
SOLUTION GRAPH:
2. $\latex \frac{d^2y}{dt^2}+4y=0$ , y(0)=2 , y’(0)=3
$\latex L{\frac{d^2y}{dt^2}}=s^2L{y(s)}-2s-3$
$\latex s^2L{y(s)}-2s-3+4L{y(s)}=0$
$\latex (s^2+4)L{y(s)}=2s+3$
$\latex L{y(s)}=\frac{2s+3}{(s^2+4)}$
the simplified laplace transform yeilds the final solution:
$\latex y(t)=2cos(2*t)+ \frac{3}{2}sin(2*t)$
SOLUTION GRAPH:
