Question

Determine the numerical solution of the differential equation
**y'+y-x=0** using the Euler and the Runge-Kutta
method until n = 5. The step size is 0.2, y(0) = 1.

No need to show calculations, I just need the summary of results
of both methods with their percent absolute error from the exact
value per y_{n}.

Abs. error will be (Exact-Approx)/Exact * 100

Answer #1

The differential equation given as dy / dx = y(x^3) -
1.4y, y (0) = 1 is calculated by taking the current h = 0.2 at the
point x = 0.6 and calculated by the Runge-Kutta method from the 4th
degree, find the relative error.
analytical solution: y(x)=e^(0.25(x^4)-1.4x)

Consider the following differential equation:
dydx=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

Use C++ in Solving Ordinary Differential Equations using
a
Fourth-Order Runge-Kutta of Your Own Creation
Assignment:
Design and construct a computer program in C++ that will
illustrate the use of a fourth-order
explicit Runge-Kutta method of your own design. In other words, you
will first have to solve the Runge-Kutta equations of condition for
the coefficients
of a fourth-order Runge-Kutta method. See the
Mathematica notebook on solving the equations for 4th order RK
method.
That notebook can be found at...

Given
(dy/dx)=(3x^3+6xy^2-x)/(2y)
with y=0.707 at x= 0, h=0.1 obtain a solution by the fourth
order Runge-Kutta method for a range x=0 to 0.5

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-Euler method

x^2y'' − 3xy'+ 4y = 0 ; y(1)=5 y'(1)=3
differential equation using the Cauchy-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.

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.

Find the general solution to the Cauchy-Euler equation:
x^2 y'' - 5xy' + 8y = 0

10.16: Write a user-defined MATLAB function that solves a
first-order ODE by applying the midpoint method (use the form of
second-order Runge-Kutta method, Eqs(10.65),(10.66)). For function
name and arguments use [x,y]=odeMIDPOINT(ODE,a,b,h,yINI). The input
argument ODE is a name for the function that calculates dy/dx. It
is a dummy name for the function that is imported into odeMIDPOINT.
The arguments a and b define the domain of the solution, h is step
size; yINI is initial value. The output arguments, x...

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