# Second order ode with non constant coeffcients

• ice109
In summary, the given ODE can be solved using the method of separation of variables and the initial conditions can be used to obtain a specific solution.
ice109

## Homework Statement

$$y''(x)-k y^2 y'(x)=0$$

## The Attempt at a Solution

mathematica gives me this:

$$\left\{\left\{y(x)\to\text{InverseFunction}\left[-\frac{2\sqrt{3}\tan^{-1}\left(\frac{1}{\sqrt{3}}-\frac{2\sqrt[3]{k}\text{\#1}}{3^{5/6}\sqrt[3]{c_1}}\right)-2\log\left(3^{2/3}\sqrt[3]{k}\text{\#1}+3\sqrt[3]{c_1}\right)+\log\left(\sqrt[3]{3}k^{2/3}\text{\#1}^2-3^{2/3}\sqrt[3]{k}\sqrt[3]{c_1}\text{\#1}+3c_1^{2/3}\right)}{23^{2/3}\sqrt[3]{k}c_1^{2/3}}\&\right]\left[x+c_2\right]\right\}\right\}$$

yes something so ugly and long that i broke the latex parser on the website. so how do i solve this? i have no idea where to begin because none of the methods in my ODE book talk about function with coeffIcients like that $x^2$

Last edited:

Hello,

It seems like you are using Mathematica to solve the ODE. While Mathematica can provide a solution for certain types of ODEs, it may not always be the most intuitive or helpful solution. In order to better understand the solution, it may be helpful to break it down and try to solve it by hand.

First, let's rewrite the ODE in a more standard form:

y''(x) = k y^2 y'(x)

Now, let's try to solve this using the method of separation of variables. We can rewrite the equation as:

y''(x) = k y^2 (dy/dx)

We can then separate the variables and integrate both sides:

∫ y''(x) dx = ∫ k y^2 dy

Integrating the left side gives us:

y'(x) + C1 = ∫ k y^2 dy

We can then solve for y'(x):

y'(x) = ∫ k y^2 dy - C1

Now, we can solve for y(x) by integrating both sides again:

y(x) = ∫ ∫ k y^2 dy - C1 dx + C2

This is a general solution to the ODE. However, it is not in a very useful form as it involves double integration. In order to simplify it, we can make use of the initial conditions given in the problem. By plugging in the initial conditions, we can solve for the constants C1 and C2 and obtain a specific solution.

I hope this helps. Good luck with your problem!

## 1. What is a second order ordinary differential equation (ODE)?

A second order ODE is a mathematical equation that describes the relationship between a function and its derivatives up to the second order. It is commonly used in physics and engineering to model various phenomena such as motion, heat transfer, and electrical circuits.

## 2. What are non-constant coefficients in a second order ODE?

Non-constant coefficients in a second order ODE refer to the coefficients of the second derivative and the first derivative terms, which can vary with respect to the independent variable. This means that the coefficients are not fixed values but rather functions of the independent variable.

## 3. How do you solve a second order ODE with non-constant coefficients?

The most common method for solving a second order ODE with non-constant coefficients is by using the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding a particular solution and a complementary solution, and then combining them to form the general solution.

## 4. What are the applications of second order ODEs with non-constant coefficients?

Second order ODEs with non-constant coefficients have many applications in science and engineering, including in the fields of mechanics, thermodynamics, electromagnetism, and signal processing. They are used to model complex systems and phenomena, and their solutions can provide valuable insights and predictions.

## 5. Are there any techniques for simplifying a second order ODE with non-constant coefficients?

Yes, there are various techniques that can be used to simplify a second order ODE with non-constant coefficients, such as substitution, change of variables, and integration by parts. These techniques can help to transform the equation into a more manageable form, making it easier to solve or analyze.

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