Question

write MATLAB scripts to solve differential equations.

Computing 1: ELE1053 Project 3E:Solving Differential Equations Project Principle Objective: Write MATLAB scripts to solve differential equations. Implementation: MatLab is an ideal environment for solving differential equations. Differential equations are a vital tool used by engineers to model, study and make predictions about the behavior of complex systems. It not only allows you to solve complex equations and systems of equations it also allows you to easily present the solutions in graphical form. You will study the analytical solutions of DEs in your Mathematics module. This project consists of two parts: 1) Follow the instructions below to construct MatLab scripts to solve a range of ordinary differential equation and systems of equations. 2) Using the techniques in part 1) as an example apply the same approach to obtain and graph the solutions of some other problems. You have already studied how to plot functions in MatLab refer to your solutions to the MatLab exercises you did during the start of the course. Part 1: Exercise 1: Solve this differential equation. dy = ty dt First, represent y by using syms to create the symbolic function y(t). syms y(t) Define the equation using == and represent differentiation using the diff function. ode = diff(y.t) ==t'y Note how you tell MatLab to code the derivatve term. Solve the equation using the 'dsolve" command, i.e. ySol(t) = dsolve(ode)

Answer 1

Exercise 1)

Code:

%Exercise 1
syms y(t)

ode = diff(y,t)==t*y;
%Showing this output
ySol(t)= dsolve(ode)

%Using boundary condition
cond = y(0)==2;

%Showing this output
ySol(t)= dsolve(ode,cond)

%Plting on a interval of [-3,+3]
fplot(ySol,[-3,3]);

PLOT:

180 160 140 120 100

Exercise 2) :

Code:

%Exercise 2
syms y(t)

ode = (diff(y,t)+y)^2==1;

%Using boundary condition
cond = y(0)==0;

%Showing this output
ySol(t)= dsolve(ode,cond)

%Plting on a interval of [-6,+10]
fplot(ySol,[-6,10]);

Plot:

Exercise 2) :

Code:

Plot:

-300 -400 6 4 -2 0 2 4 6 8 10

Exercise 3) :

Code:

%Exercise 3
syms y(x)
%1st Differential
Dy = diff(y);

%original equation
ode = diff(y,x,2)==cos(2*x)-y;

%Using boundary condition
cond1 = y(0)==1;
cond2 = Dy(0)==0;

conds = [cond1,cond2];

%Solving equation using 'conds' as condition
ySol(x)= dsolve(ode,conds)

ySol = simplify(ySol)
%Polting on an assumed interval of [-6,+10]
fplot(ySol,[-6,10]);

Plot:

0.5 -0.5 | 6 4 20 24 68 10

Exercise 4) :

Solution 1

uSol(t) =

C10*cos(4*t)*exp(3*t) + C9*sin(4*t)*exp(3*t)


vSol(t) =

C9*cos(4*t)*exp(3*t) - C10*sin(4*t)*exp(3*t)

Code:

%Exercise 4
syms u(t) v(t)
%1st Differential
ode1 = diff(u)==3*u+4*v;
%2nd Differential
ode2 = diff(v) == -4*u+3*v;
%original equation
odes = [ode1,ode2];
%Solving Without boundary condition
[uSol(t),vSol(t)]=dsolve(odes)

%Using boundary condition
cond1 = u(0)==0;
cond2 = v(0)==1;

conds = [cond1,cond2];

%Solving equation using 'conds' as condition
[uSol(t),vSol(t)]= dsolve(odes,conds)

%Polting with default range
fplot(uSol);
hold on
fplot(vSol);
grid on
legend('uSol','vSol','Location','best')

Plot:

3,106 USol v Sol 1.5 -0.5 L -5 4 -3 -2 -1 0 1 2 3 4 5

Problem 1) :

Solution:

ySol(x) =

(5*cos(x))/3 + sin(x)*(sin(3*x)/6 + sin(x)/2) - (2*cos(x)*(6*tan(x/2)^2 - 3*tan(x/2)^4 + 1))/(3*(tan(x/2)^2 + 1)^3) - (sin(x)*(6*cos(2) + 9*sin(2)^2*tan(1)^2 + 9*sin(2)^2*tan(1)^4 + 3*sin(2)^2*tan(1)^6 + sin(2)*sin(6) + 6*cos(2)*tan(1)^2 + 42*cos(2)*tan(1)^4 + 10*cos(2)*tan(1)^6 + 3*sin(2)^2 - 90*tan(1)^2 - 90*tan(1)^4 - 30*tan(1)^6 + 3*sin(2)*sin(6)*tan(1)^2 + 3*sin(2)*sin(6)*tan(1)^4 + sin(2)*sin(6)*tan(1)^6 - 30))/(6*sin(2)*(tan(1)^2 + 1)^3)

Code:

%Problem 1
syms y(x)
% Differential
Dy = diff(y);
%original Differential
ode = diff(y,x,2) == cos(2*x)-y;

%Using boundary condition
cond1 = y(0)==1;
cond2 = y(2)==5;

conds = [cond1,cond2];

%Solving equation using 'conds' as condition
ySol(x) = dsolve(ode,conds)

%Polting with default range
fplot(ySol);

Plot:

5 4 3 2 1 0 1 2 3 4 5

Problem 2) :

SOlution:

uSol(x) =

(pi*exp(x))/3 - exp(-x/2)*cos((3^(1/2)*x)/2)*(pi/3 - 1) - (3^(1/2)*exp(-x/2)*sin((3^(1/2)*x)/2)*(pi + 1))/3

Code:

%Problem 2
syms u(x)
% Differential
Du1 = diff(u);
Du2 = diff(Du1);
%original Differential
ode = diff(u,x,3) == u;

%Using boundary condition
cond1 = u(0)==1;
cond2 = Du1(0)==-1;
cond3 = Du2(0) ==pi;

conds = [cond1,cond2,cond3];

%Solving equation using 'conds' as condition
uSol(x) = dsolve(ode,conds)

%Polting with default range
fplot(uSol);

Plot:

-5 -4 -3 -2 -1 0 1 2 3 4

Problem 3) :

SOlution:

xSol(t) =

2^(1/2)*((t*(t^2 - 3))/3)^(1/2) - 2
- 2^(1/2)*((t*(t^2 - 3))/3)^(1/2) - 2

Code:

%Problem 3
syms x(t)

%original Differential
ode = diff(x,t) == (t^2-1)/(x+2);

%Using boundary condition
cond = x(0)==-2;

%Solving equation using 'cond' as condition
xSol(t) = dsolve(ode,cond)

%Polting with default range
fplot(xSol);

Plot:

s_ _ £_ _ - -- ----- -- --- - -- ----- -- --- - 。 :

Problem 4) :

SOlution:

xSol(t) =

3^(1/3)*(4*t*(log(t) - 1) + 4)^(1/3)
3^(1/3)*((3^(1/2)*1i)/2 - 1/2)*(4*t*(log(t) - 1) + 4)^(1/3)
-3^(1/3)*((3^(1/2)*1i)/2 + 1/2)*(4*t*(log(t) - 1) + 4)^(1/3)

Code:

%Problem 4
syms x(t)

%original Differential
ode = diff(x,t) == 4*log(t)/x^2;

%Using boundary condition
cond = x(1)==0;

%Solving equation using 'cond' as condition
xSol(t) = dsolve(ode,cond)

%Polting with default range
fplot(xSol);

Plot:

- - - - - - - - - - - : -- - ター 1 .5 2 2.5_ 3_ 3.5 4 4.5 5

Problem 5) :

Solution:

xSol(t) =

4^(1/4)*t*(log(t) + 64)^(1/4)

Code:

%Problem 5
syms x(t)

%original Differential
ode = (x^3)*t*diff(x,t) == t^4+x^4;

%Using boundary condition
cond = x(1)==4;

%Solving equation using 'cond' as condition
xSol(t) = dsolve(ode,cond)

%Polting with default range
fplot(xSol);

Plot:

10.5 1 1.5 2 2.5 3 3.5 4 4. 5 5

Problem 6) :

EQUATION IS VERY BLURRY. UNABLE TO COMPREHEND WHAT'S WRITTEN IN EXPONENTIAL.

Kindly Comment correct Equation, I will give you the code for it.

Problem 7) :

UNABLE TO UNDERSTAND WHAT'S WRITTEN IN THE EXPONENTIAL.

USING THIS GUESSED EQUATION:

da 1– 2x -= 4ttet dtt z(1) = 0

Solution:

xSol(t) =

(2*exp(t) + t^2*exp(t) - 2*t*exp(t) + t^2/2 + t^4)/t^2 - (exp(1) + 3/2)/t^2

Code:

%Problem 7
syms x(t)

%original Differential
ode = diff(x,t)-(1-2*x)/(t) == 4*t+exp(t);

%Using boundary condition
cond = x(1)==0;

%Solving equation using 'cond' as condition
xSol(t) = dsolve(ode,cond)

%Polting with default range
fplot(xSol);

Plot:

- - - - - - - - - - - - - - - - - - - - - - - - - - 432 10 1 2 3 --150 --200F

Problem 8) :

Solution:

xSol(t) =

exp(t/2)*cos((5^(1/2)*t)/2) - (5^(1/2)*exp(t/2)*sin((5^(1/2)*t)/2))/5

Code:

%Problem 8
syms x(t)
%Differntial 1
Dx =diff(x);
%original Differential
ode = 2*diff(x,t,2)-2*Dx +3*x==0;

%Using boundary condition
cond1 = x(0)==1;
cond2 = Dx(0)==0;

conds=[cond1,cond2];

%Solving equation using 'conds' as condition
xSol(t) = dsolve(ode,conds)

%Polting with default range
fplot(xSol);

Plot:

| 5 4 3 2 4 0 1 2 3 4 5

Problem 9) :

Solution:

xSol(t) =

2*exp(-2*t)*exp(2) - 5*t*exp(-2*t)*exp(2) + 4*t^2*exp(-2*t)*exp(2)

Code:

%Problem 9
syms x(t)
%Differntial 1
Dx1 =diff(x);
Dx2 = diff(Dx1);
%original Differential
ode = diff(x,t,3)+6*Dx2 +12*Dx1+8*x==0;

%Using boundary condition
cond1 = x(1)==1;
cond2 = Dx1(1)==1;
cond3 = Dx2(1)==0;
conds=[cond1,cond2,cond3];

%Solving equation using 'conds' as condition
xSol(t) = dsolve(ode,conds)

%Polting with default range
fplot(xSol);

Plot:

X 107 5 4 -3 -2 -1 0 1 2 3 4 5

Problem 10) :

Solution:

xSol(t) =

(12*exp(-t))/25 - (4*cos(2*t + atan(4/3)))/5 + t*exp(-t)*((4*cos(atan(4/3) + 2)*exp(1))/5 - 12/25)

Code:

%Problem 10
syms x(t)
%Differntial 1
Dx =diff(x);

%original Differential
ode = diff(x,t,2)+2*Dx +x==4*cos(2*t);

%Using boundary condition
cond1 = x(0)==0;
cond2 = Dx(0)==2;

conds=[cond1,cond];

%Solving equation using 'conds' as condition
xSol(t) = dsolve(ode,conds)

%Polting with default range
fplot(xSol);

Plot:

2000 1800 1600 1400 1200 1000 아 800 600 400 아 200 1 5 4 3 2 -1 0 1 2 3 4 5

Problem 11) :

Solution:

xSol(t) =

(8*exp(-t))/5 - (44*cos(3*t)*exp(-2*t))/65 + (27*sin(3*t)*exp(-2*t))/65 + 1/13

Code:

%Problem 11
syms x(t)
%Differntial 1
Dx1 =diff(x);
Dx2 = diff(Dx1);
%original Differential
ode = diff(x,t,3)+5*Dx2 +17*Dx1+13*x==1;

%Using boundary condition
cond1 = x(0)==1;
cond2 = Dx1(0)==1;
cond3 = Dx2(0)==0;
conds=[cond1,cond2,cond3];

%Solving equation using 'conds' as condition
xSol(t) = dsolve(ode,conds)

%Polting with default range
fplot(xSol);

Plot:

4000 上 2000 -2000 -4000 -6000 - 5 4 -3 -2 - 1 0 1 2 3 4 5

Problem 12) :

Solution:

xSol(t) =

sin((7^(1/2)*t)/2)*(cos(4*t - (7^(1/2)*t)/2)/48 - cos(4*t + (7^(1/2)*t)/2)/48 + sin(4*t - (7^(1/2)*t)/2)/16 - sin(4*t + (7^(1/2)*t)/2)/16 + (17*7^(1/2)*cos(4*t - (7^(1/2)*t)/2))/336 + (17*7^(1/2)*cos(4*t + (7^(1/2)*t)/2))/336 + (7^(1/2)*sin(4*t - (7^(1/2)*t)/2))/16 + (7^(1/2)*sin(4*t + (7^(1/2)*t)/2))/16) - cos((7^(1/2)*t)/2)*(cos(4*t - (7^(1/2)*t)/2)/16 + cos(4*t + (7^(1/2)*t)/2)/16 - sin(4*t - (7^(1/2)*t)/2)/48 - sin(4*t + (7^(1/2)*t)/2)/48 + (7^(1/2)*cos(4*t - (7^(1/2)*t)/2))/16 - (7^(1/2)*cos(4*t + (7^(1/2)*t)/2))/16 - (17*7^(1/2)*sin(4*t - (7^(1/2)*t)/2))/336 + (17*7^(1/2)*sin(4*t + (7^(1/2)*t)/2))/336) + C3*exp((3*t)/2)*cos((7^(1/2)*t)/2) - C4*exp((3*t)/2)*sin((7^(1/2)*t)/2)

Code:

%Problem 12
syms x(t)
%Differntial 1
Dx =diff(x);

%original Differential
ode = diff(x,t,2)-3*Dx+4*x==cos(4*t)-2*sin(4*t);

%Solving equation without Conditions
xSol(t) = dsolve(ode)