In: Advanced Math

Solve this differential equation using
**Matlab**

yy' + xy^{2} =x , with y(0)=5 for x=0 to 2.5 with a step
size 0.25

(a) Analytical

(b) Euler

(c) Heun

d) 4th order R-K method

**Display all results on the same graph**

**MATLAB
Code:**

close all

clear

clc

dydx = @(x,y) (x - x*y^2)/y; % Given ODE, y'

h = 0.25; % Step size

xi = 0; xf = 2.5;

x = xi:h:xf;

n = (xf - xi)/h; % Number of nodes

% Part (a)

% Analytical solution

syms y_exact(x_exact)

eqn = y_exact*diff(y_exact, x_exact) + x_exact*y_exact^2 ==
x_exact; % ODE

cond = y_exact(0) == 5; % Initial condition

y_exact = dsolve(eqn, cond); % Solve for exact solution

x_vals = linspace(xi, xf); % Bunch of new samples for a smoother
plot

plot(x_vals, subs(y_exact, x_vals)), hold on

% Part (b)

% Euler's Method

y_euler(1) = 5; % Initial condition

for i=1:n

m = dydx(x(i), y_euler(i));

y_euler(i+1) = y_euler(i) + h*m; % Euler's Update

end

plot(x, y_euler, 'o-')

% Part (c)

y_heun(1) = 5; % Initial condition

for i = 1:n

m1 = dydx(x(i), y_heun(i));

m2 = dydx(x(i) + h, y_heun(i) + h*m1);

y_heun(i+1) = y_heun(i) + 0.5*h*(m1 + m2); % Heun's Update

end

plot(x, y_heun, '*-')

% Part (d)

% RK4 Method

y_rk4(1) = 5; % Initial condition

for i = 1:n

k1 = dydx(x(i), y_rk4(i));

k2 = dydx(x(i) + 0.5*h, y_rk4(i) + 0.5*h*k1);

k3 = dydx(x(i) + 0.5*h, y_rk4(i) + 0.5*h*k2);

k4 = dydx(x(i) + h, y_rk4(i) + k3*h);

y_rk4(i + 1) = y_rk4(i) + (1/6)*(k1 + 2*k2 + 2*k3 + k4)*h; % RK4
Update

end

plot(x, y_rk4, 's-')

xlabel('x'), ylabel('y(x)'),

title('Solution of ODE')

legend('Exact Solution', 'Euler', 'Heun', 'RK4')

**Plot:**

Consider the differential equation:
y'(x)+3xy+y^2=0.
y(1)=0. h=0.1
Solve the differential equation to determine y(1.3)
using:
a. Euler Method
b. Second order Taylor series method
c. Second order Runge Kutta method
d. Fourth order Runge-Kutta method
e. Heun’s predictor corrector method
f. Midpoint method

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y''-0.25y=3sin0.5t where y(0)=0 and y'(0)=0

Solve the differential equation
1. a) 2xy"+ y' + y = 0
b) (x-1)y'' + 3y = 0

Solve the following differential equation, using laplace
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y''+ty-y=0
where
y(0)=0
and
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(Answer using fractions)

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