Question

Consider the following differential equation:
*dy**dx**=x+y*

With initial condition: y = 1 when x = 0

- Using the Euler forward method, solve this differential equation for the range x = 0 to x = 0.5 in increments (step) of 0.1
- Check that the theoretical solution is y(x) = - x -1 , Find the error between the theoretical solution and the solution given by Euler method at x = 0.1 and x = 0.5 , correct to three decimal places

Answer #1

a)

code:

#include<stdio.h>

float fun(float x,float y)

{

float f;

f=x+y;

return f;

}

main()

{

float a,b,x,y,h,t,k;

printf("\nEnter x0,y0,h,xn: ");

scanf("%f%f%f%f",&a,&b,&h,&t);

x=a;

y=b;

printf("\n x\t y\n");

while(x<t)

{

k=h*fun(x,y);

y=y+k;

x=x+h;

printf("%0.3f\t%0.3f\n",x,y);

}

}

result:

b)

Given the second-order differential equation
y''(x) − xy'(x) + x^2 y(x) = 0
with initial conditions
y(0) = 0, y'(0) = 1.
(a) Write this equation as a system of 2 first order
differential equations.
(b) Approximate its solution by using the forward Euler
method.

(1 point) A Bernoulli differential equation is one of the
form
dydx+P(x)y=Q(x)yn (∗)
Observe that, if n=0 or 1, the Bernoulli equation is linear. For
other values of n, the substitution u=y1−n transforms the Bernoulli
equation into the linear equation
dudx+(1−n)P(x)u=(1−n)Q(x).dudx+(1−n)P(x)u=(1−n)Q(x).
Consider the initial value problem
y′=−y(1+9xy3), y(0)=−3.
(a) This differential equation can be written in the form (∗)
with
P(x)= ,
Q(x)= , and
n=.
(b) The substitution u= will transform it into the linear
equation
dudx+ u= .
(c) Using...

1) Basic Euler’s Method:
y'+xysin/y+1 y(0)=1
a) What is the initial condition?
b) What order is this differential equation?
c) Is this an autonomous differential equation?
d) Is this a separable differential equation?
e) Find the general solution to the given differential equation,
by hand. You will not be able to completely solve for y(x) – that’s
ok. Write out all your work and attach it to your Questions
tab.
f) Using the initial condition, solve the initial value problem...

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please write as neat as possible better if typed and explain
clearly with step by step work

Solve the Initial Value Problem:
a) dydx+2y=9, y(0)=0 y(x)=_______________
b) dydx+ycosx=5cosx,
y(0)=7d y(x)=______________
c) Find the general solution, y(t), which solves the problem
below, by the method of integrating factors.
8t dy/dt +y=t^3, t>0
Put the problem in standard form.
Then find the integrating factor, μ(t)= ,__________
and finally find y(t)= __________ . (use C as the unkown
constant.)
d) Solve the following initial value problem:
t dy/dt+6y=7t
with y(1)=2
Put the problem in standard form.
Then find the integrating...

Solve the Homogeneous differential equation
(7 y^2 + 1 xy)dx - 1 x^2 dy = 0
(a) A one-parameter family of solution of the equation is y(x)
=
(b) The particular solution of the equation subject to the
initial condition y(1) =1/7.

Consider the following initial value problems and solve
by hand.
y' = 2x-3y+1
y(1) = 5
Let the step size 0.1.
Then find the approximate value of y(1.2) by using Euler
Method and Improved Euler Method.
(Please show your work for both the Euler Method and the
Improved Euler Method.)

y’ – y = 2x -1 y(0) = 1 , 0 ≤ x ≤ 0.2
Use the Euler method to solve the following initial value
problem
(a) Check whether the function y = 2 ex -2x-
1 is the analytical solution ;
(b) Find the errors by comparing the exact values you’re your
numerical results (h = 0.05 and h = 0.1)
and Discuss the issue of numerical stability.

Consider the differential equation y′′+ 9y′= 0.(
a) Let u=y′=dy/dt. Rewrite the differential equation as a
first-order differential equation in terms of the variables u.
Solve the first-order differential equation for u (using either
separation of variables or an integrating factor) and integrate u
to find y.
(b) Write out the auxiliary equation for the differential
equation and use the methods of Section 4.2/4.3 to find the general
solution.
(c) Find the solution to the initial value problem y′′+ 9y′=...

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