Nonhomogeneous 2nd order dif equation

In summary, the conversation is about solving a nonhomogeneous second-order differential equation using the constant variation method or other methods. The equation is xy''-2y' = -\frac{2}{x^2}, and the initial conditions are y(1)=-\frac{1}{3} and y'(1)=1. The speaker also mentions changing variables to solve the equation.
  • #1
darjiaus7
1
0
Hello guys . I want to do nonhomogeneous 2nd order dif equation..I am trying to do this for 2 days , but I can't get good answer. Can you show me how to do this equation with constant variation method ( i know it best) or other I would be very gratefull , because after 2 days of trying I am surrendered... [itex]xy''-2y' = -\frac{2}{x^2} ; y(1)=-\frac{1}{3} ; y'(1)=1[/itex]
 
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  • #2
darjiaus7 said:
Hello guys . I want to do nonhomogeneous 2nd order dif equation..I am trying to do this for 2 days , but I can't get good answer. Can you show me how to do this equation with constant variation method ( i know it best) or other I would be very gratefull , because after 2 days of trying I am surrendered... [itex]xy''-2y' = -\frac{2}{x^2} ; y(1)=-\frac{1}{3} ; y'(1)=1[/itex]

This is a first-order ODE for [itex]u = y'[/itex]: [tex]
u' -\frac{2}{x}u = -\frac{2}{x^3}.[/tex] It has an integrating factor.
 
  • #3
change variables
##\mathrm{u}(x)=x^{-2} \mathrm{y}^\prime (x)##
 

1. What is a nonhomogeneous 2nd order differential equation?

A nonhomogeneous 2nd order differential equation is a mathematical equation that involves a second derivative of a variable with respect to another variable, with an additional non-zero term known as the forcing function. This forcing function makes the equation nonhomogeneous, as opposed to a homogeneous 2nd order differential equation where the forcing function is equal to zero.

2. How do you solve a nonhomogeneous 2nd order differential equation?

To solve a nonhomogeneous 2nd order differential equation, you can use the method of undetermined coefficients or variation of parameters. In the method of undetermined coefficients, you assume a particular solution and solve for the coefficients. In variation of parameters, you find a general solution and then use a variation of the parameters to find a particular solution.

3. What is the general solution of a nonhomogeneous 2nd order differential equation?

The general solution of a nonhomogeneous 2nd order differential equation is the sum of the complementary function (solution to the corresponding homogeneous equation) and the particular solution (solution to the nonhomogeneous equation). This general solution satisfies both the differential equation and any initial conditions given.

4. Can a nonhomogeneous 2nd order differential equation have multiple solutions?

Yes, a nonhomogeneous 2nd order differential equation can have multiple solutions. This is because the nonhomogeneous term allows for the addition of various particular solutions to the complementary function, resulting in multiple solutions that all satisfy the differential equation.

5. What are the applications of nonhomogeneous 2nd order differential equations?

Nonhomogeneous 2nd order differential equations have many applications in physics, engineering, and other areas of science. They can be used to model real-world phenomena such as oscillating systems, electrical circuits, and heat transfer. They are also used in the study of vibrations, control systems, and quantum mechanics.

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