Second Order Linear Differential Equation (Non-constant coefficients)

In summary, the conversation discusses finding the general solution for a differential equation involving a squared term. The individual has concerns about this term causing the equation to be non-linear, but it is suggested to use separation of variables to solve it.
  • #1
SherlockOhms
310
0

Homework Statement


Find the general solution for the following differential equation:
y'' + x(y')^2 = 0.


Homework Equations


Integration, differentiation...


The Attempt at a Solution


Usually for these sort of DE you could use the substitution v(x) = y'(x) and this would simplify such an equation to a first order DE. The (y')^2 part is throwing me off as this would give the equation:
v(x)' + x(v(x))^2 = 0.
This would be grand if it weren't for the v(x)^2 term. Any ideas on how to get around this? Thanks.
 
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  • #2
Are you worried about the v^2 term because it makes the eqn non-linear? If so, there are ways to solve such an equation.
 
  • #3
Yeah. We've only covered linear DE's.
 
  • #4
DAPOS said:
Yeah. We've only covered linear DE's.

Are you at all familiar with separation of variables?
 
  • #5
Yeah. So I'll have v'(x) = (-x)(v(x))^2 and I can then separate the variables and solve. Thanks!
 

1. What is a second order linear differential equation with non-constant coefficients?

A second order linear differential equation with non-constant coefficients is a mathematical equation that involves a second derivative of a dependent variable with respect to an independent variable, as well as first derivative and constant coefficients. It can be written in the general form of y'' + p(x)y' + q(x)y = r(x), where p(x) and q(x) are non-constant functions of x.

2. How do you solve a second order linear differential equation with non-constant coefficients?

To solve a second order linear differential equation with non-constant coefficients, you can use the method of undetermined coefficients or variation of parameters. In the method of undetermined coefficients, you assume a particular solution based on the form of the non-homogeneous term and solve for the unknown coefficients. In variation of parameters, you use a variation of the general solution of the corresponding homogeneous equation to find a particular solution. You can also solve these equations using numerical methods such as Euler's method or Runge-Kutta method.

3. What is the difference between a homogeneous and non-homogeneous second order linear differential equation?

A homogeneous second order linear differential equation has a non-zero non-homogeneous term, while a homogeneous equation has a zero non-homogeneous term. This means that the solutions of a homogeneous equation will always satisfy the equation, while the solutions of a non-homogeneous equation may not satisfy it.

4. Can a second order linear differential equation with non-constant coefficients have complex solutions?

Yes, a second order linear differential equation with non-constant coefficients can have complex solutions. This is because the coefficients and variables in the equation can be complex numbers, and the solutions can involve complex functions such as sine and cosine. Complex solutions can also arise when using the method of undetermined coefficients to solve non-homogeneous equations.

5. What are some real-world applications of second order linear differential equations with non-constant coefficients?

Second order linear differential equations with non-constant coefficients have many real-world applications in physics, engineering, and other fields. They can be used to model systems that involve forces, electricity and magnetism, heat transfer, and many other phenomena. For example, they can be used to model the motion of a spring-mass system, the decay of a radioactive substance, or the behavior of an electrical circuit.

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