This document discusses using MATLAB to solve differential equations through Laplace transformations. It introduces key terms like the Laplace operator and generating function. It then demonstrates how to use MATLAB commands like "laplace" and "ilaplace" to calculate the Laplace transform of a function and take the inverse Laplace transform. Examples are provided, such as finding the Laplace transform of the function f(t)=-1.25+3.5t*exp(-2t)+1.25*exp(-2t).

1. Differential equation &
LAPLACE TRANSFORmation
with MATLAB
RAVI JINDAL
Joint Masters, SEGE (M1) Second semester
B.K. Birla institute of Engineering & Technology, Pilani

2. Differential Equations with
MATLAB
 MATLAB has some powerful features for
solving differential equations of all types. We
will explore some of these features for the
Constant Coefficient Linear Ordinary
Differential Equation forms.
 The approach here will be that of the Symbolic
Math Toolbox. The result will be the form of
the function and it may be readily plotted with
MATLAB.

3. Symbolic Differential Equation
Terms
2
2
n
n
y
dy
dt
d y
dt
d y
dt
y
Dy
D2y
Dny

4. Finding Solutions to Differential
Equations
 Solving a First Order Differential Equation
 Solving a Second Order Differential
Equation
 Solving Simultaneous Differential
Equations
 Solving Nonlinear Differential Equations
 Numerical Solution of a Differential
Equation

5. Solving a 1st Order DE
 Consider the differential
equation:
122 =+ y
dt
dy
The general solution is given by:
The Matlab command used to solve differential
equations is dsolve .
Verify the solution using dsolve command

6. Solving a Differential
Equation in Matlab
C1 is a constant which is specified by way of the
initial condition
Dy means dy/dt and D2y means d2y/dt2 etc
» syms y t
» ys=dsolve('Dy+2*y=12')
ys =6+exp(-2*t)*C1

7. Verify Results
 Verify results given y(0) = 9
» ys=dsolve('Dy+2*y=12','y(0)=9')
ys =
6+3*exp(-2*t)
39)0( 1 =→= Cy

8. Solving a 2nd Order DE
8
 Find the general solution of:
02
2
2
=+ yc
dt
yd
)cos()sin()( 21 ctCctCty +=
» syms c y
» ys=dsolve('D2y = - c^2*y')
ys = C1*sin(c*t)+C2*cos (c*t)

9. 9
 Solve the following set of differential equations:
Solving Simultaneous
Differential Equations Example
yx
dt
dx
43 += yx
dt
dy
34 +−=
Syntax for solving simultaneous differential equations is:
dsolve('equ1', 'equ2',…)

10.  The general solution is given by:
General Solution
)4sin()4cos()( 3
2
3
1 tectectx tt
+=
)4cos()4sin()( 3
2
3
1 tectecty tt
+−=
yx
dt
dx
43 += yx
dt
dy
34 +−=
Given the equations:

11. Matlab Verification
» syms x y t
» [x,y]=dsolve('Dx=3*x+4*y','Dy=-4*x+3*y')
x = exp(3*t)*(cos(4*t)*C1+sin(4*t)*C2)
y = -exp(3*t)*(sin(4*t)*C1-cos(4*t)*C2)
yx
dt
dx
43 += yx
dt
dy
34 +−=
Given the
equations:
 General
solution is:
)4sin()4cos()( 3
2
3
1 tectectx tt
+=
)4cos()4sin()( 3
2
3
1 tectecty tt
+−=

12.  Solve the previous system with the initial conditions:
Initial Conditions
0)0( =x 1)0( =y
» [x,y]=dsolve('Dx=3*x+4*y','Dy=-4*x+3*y',
'y(0)=1','x(0)=0')
x = exp(3*t)*sin(4*t)
y = exp(3*t)*cos(4*t) )4cos(
)4sin(
3
3
tey
tex
t
t
=
=

13. Non-Linear Differential Equation Example
 Solve the differential equation: 2
4 y
dt
dy
−=
Subject to initial condition:
1)0( =y
» syms y t
» y=dsolve('Dy=4-y^2','y(0)=1')
» y=simplify(y)
y =
2*(3*exp(4*t)-1)/(1+3*exp(4*t))
( )
t
t
e
e
ty 4
4
31
132
)(
+
−
=

14.  If another independent variable, other than t, is used, it must
be introduced in the dsolve command
Specifying the Independent Parameter of a
Differential Equation
122 =+ y
dx
dy
» y=dsolve('Dy+2*y=12','x')
y = 6+exp(-2*x)*C1
Solve the differential equation:
x
eCxy 2
16)( −
+=

15. Numerical Solution Example
 Not all non-linear differential equations have a closed
form solution, but a numerical solution can be found
Solve the differential equation:
Subject to initial conditions:
0)sin(92
2
=+ y
dt
yd
1)0( =y
0)0( =
•
y

16. 16
Rewrite Differential Equation
yx =1
••
== 12 xyx
)sin(9
)sin(9
12
2
xx
yyx
−=
−==
•
•••
0)sin(92
2
=+ y
dt
yd
1)0()0(1 == yx
0)0()0(2 ==
•
yx
Rewrite in the
following form
)sin(92
2
yy
dt
yd
−==
••

17. 17
Solve DE with MATLAB.
>> y = dsolve ('D2y + 3*Dy + 2*y = 24',
'y(0)=10', 'Dy(0)=0')
y = 12+2*exp(-2*t)-4*exp(-t)
>> ezplot(y, [0 6])
2
2
3 2 24
d y dy
y
dt dt
+ + =
(0) 10y = '(0) 0y =

19. Definition of Laplace
Transformation:
Let f(t) be a given function defined for all t ≥ 0 ,
then the Laplace Transformation of f(t)
is defined as
Here,
L = Laplace Transform Operator.
f(t) =determining function, depends on t .
F(s)= Generating function, depends on s .

20. Differential
equations
Input
excitation e(t)
Output
response r(t)
Time Domain Frequency Domain
Algebraic
equations
Input
excitation E(s)
Output
response R(s)
Laplace Transform
Inverse Laplace Transform
The Laplace Transformation

21. Laplace Transforms with MATLAB
Calculating the Laplace F(s) transform of a function f(t) is
quite simple in Matlab . First you need to specify that the
variable t and s are symbolic ones. This is done with the
command
>> syms t s
The actual command to calculate the transform is
>> F = Laplace (f , t , s)

22. example for the function f(t)
>> syms t s
>> f=-1.25+3.5*t*exp(-2*t)+1.25*exp(-2*t);
>> F = laplace ( f , t , s)
F = -5/4/s+7/2/(s+2)^2+5/4/(s+2)
>> simplify(F)
ans = (s-5)/s/(s+2)^2
>> pretty (ans)

23. Inverse Laplace Transform
The command one uses now is ilaplace .
>> syms t s
>> F=(s-5)/(s*(s+2)^2);
>> ilaplace(F)
ans = -5/4+(7/2*t+5/4)*exp(-2*t)
>> simplify(ans)
ans = -5/4+7/2*t*exp(-2*t)+5/4*exp(-2*t)
>> pretty(ans)
- 5/4 + 7/2 t exp(-2 t) + 5/4 exp(-2 t)

24. Reference
 http://www.mathworks.in/help/symbolic/simpli
 https://www.google.co.in/#q=laplace+transform+

25. Thank You 