Question

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.

Answer #1

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

Power series
Find the particular solution of the differential equation:
(x^2+1)y"+xy'-4y=0 given the boundary conditions x=0, y=1 and y'=1.
Use only the 7th degree term of the solution. Solve for y at x=2.
Write your answer in whole number.

Consider the differential equation x2y''+xy'-y=0,
x>0.
a. Verify that y(x)=x is a solution.
b. Find a second linearly independent solution using the method
of reduction of order. [Please use y2(x) =
v(x)y1(x)]

For the below ordinary differential equation with initial
conditions, state the order and determine if the equation is linear
or nonlinear. Then find the solution of the ordinary differential
equation, and apply the initial conditions. Verify your solution.
x^2/(y^2-1) dy/dx=(3x^3)/y, y(0)=2

For the below ordinary differential equation with initial
conditions, state the order and determine if the equation is linear
or nonlinear. Then find the solution of the ordinary differential
equation, and apply the initial conditions. Verify your
solution.
y''-3y'+2y=0; y(0)=1,y' (0)=2.

1. Find the general solution to the differential equation y''+
xy' + x^2 y = 0 using power series techniques

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.

solve differential equation ((x)2 - xy +(y)2)dx - xydy
= 0
solve differential equation (x^2-xy+y^2)dx - xydy =
0

Find a second solution of the given differential equation. Use
reductionof order or Formula (4). Assume an appropriate interval of
validity.
(1 +x)y′′+xy′−y= 0 ; y1=x

Use the method for solving homogeneous equations to solve the
following differential equation.
(9x^2-y^2)dx+(xy-x^3y^-1)dy=0
solution is F(x,y)=C, Where C= ?

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