# Second-order ODE, reduction of order?

• Combinatus
In summary, the conversation discusses finding a specified particular solution for the equation (x^2+2y')y'' + 2xy' = 0, with initial conditions y(0)=1 and y'(0)=0. The method of substitution is suggested, but it results in a cumbersome expression that may be difficult to integrate. A potential solution is mentioned, but the reply points out that the initial condition for the substituted variable is incorrect.
Combinatus

## Homework Statement

Find the specified particular solution:

$(x^2+2y')y'' + 2xy' = 0, y(0)=1, y'(0)=0$

## The Attempt at a Solution

The equation seems amenable to the substitution $p=y'$, so it can be transformed into $(x^2 + 2p)p' + 2xp=0$, or $(x^2 + 2p)dp + 2xpdp=0$. Since $\frac{\partial (x^2+2p)}{\partial x} = 2x = \frac{\partial (2xp)}{\partial p}$, the latter equation is exact, and is solvable through the method of exact equations. Unfortunately, this yields the rather cumbersome expression $p=\frac{-x^2 \pm \sqrt{k+x^4}}{2}$ for some constant $k$, which I would have a hard time integrating, I think.

Any clever tricks?

Hi Combinatus!

erm … p(0) = 0 !

tiny-tim said:
Hi Combinatus!

erm … p(0) = 0 !
Yes, I just noticed that after looking at my scribbles, and hoped no-one would have had time to reply yet.

Oh well. Thanks.

Last edited:

## 1) What is a second-order ODE?

A second-order ODE (ordinary differential equation) is a mathematical equation that relates a function and its derivatives up to the second order. It is commonly used to model physical phenomena in fields such as physics, engineering, and economics.

## 2) What is reduction of order in a second-order ODE?

Reduction of order is a technique used to solve a second-order ODE by reducing it to a first-order ODE. This is done by making a substitution that reduces the order of the equation, allowing for easier integration and solution.

## 3) When is reduction of order used in solving a second-order ODE?

Reduction of order is typically used when one solution to the second-order ODE is known, but the second solution is unknown. It is also used when the second-order ODE is difficult to solve directly, and reduction of order simplifies the problem.

## 4) What is the general process for reduction of order in a second-order ODE?

The general process for reduction of order involves making a substitution u=y'/y, which reduces the second-order ODE to a first-order ODE. Then, the first-order ODE is solved using standard techniques such as separation of variables or integrating factors. Finally, the original second-order ODE is solved using the solution of the first-order ODE.

## 5) What are some common applications of second-order ODEs and reduction of order?

Second-order ODEs and reduction of order are used in many fields, including physics, engineering, economics, and biology. They are commonly used to model systems such as oscillating springs, electrical circuits, population growth, and chemical reactions. Reduction of order is also a useful technique in solving differential equations in general, making it an important tool for scientists and engineers.

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