# Help solving non-linear second order DE

• Pete69
In summary, the conversation is about solving a second order differential equation with the form x'' + ax' + bx^n = 0, where x^n means x to the power n, and a and b are constants. The person asking for help has tried substituting x'=u to get a first order linear equation, but got lost in the algebra. They are asking for a different method to solve the equation. Another person suggests solving the associated quadratic and solving the linear equation x'' + ax' = 0 first. The conversation also touches on the difficulty of solving non-linear differential equations, and the blind leading the blind. It is revealed that the problem comes from a course, but the person asking for help does not have experience with
Pete69
could anyone help me with solving this second order differential equation? I am a noob on here so not sure how you get the mathplayer stuff on... so the equation is of the form

x'' +ax' + bx^n = 0

(x^n means x to the power n, with a and b constants). i tried substituting x'=u to get a first order linear, but then got lost in the algebra of solving the linear DE with a integrating factor.. so is there a different method?

thanks, Pete

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Don't you solve the associated quadratic?

well substituing u=x' i get the equation

u' + au = -bx^n

which is of the form

y' +ay = f(x)

and so I multiply by the relevant integrating factor to solve, but the algebra gets very hard..

i only know how to solve linear second order DE, and got the idea of substituting x'=u from other posts, and so don't know if it is the correct way to go about this problem...

could you explain what the associated quadratic you are talking about is, as its probably something i haven't yet come across...

thanks, Pete

Don't you just have

x'' + ax' = -bx^n

So first solve x'' + ax' = 0, do you know how to do that?

Then assume a solution to solve the = -bx^n part?

aah yes coz the sums of the solutions is a solution or summit like tht right??

cheers

wait... i don't kno how to solve x'' + ax' = 0 haha

Find the roots of q^2 + a*q = 0

I'm also not sure why you think this is a nonlinear DE? I googled for "solving second order linear DEs", this is one of the links you might want to read: http://silmaril.math.sci.qut.edu.au/~gustafso/mab112/topic12/

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wel the only second order DE i have experience of solving are linear constant coefficient ones, and this doesn't look like any I've come across before.. so i jst assumed this was non-linear... thanks for the link and help, i think i get it now..

ive followed the link, and have come to a solution of

y = A + B*exp(-ax) (1) for y'' + ay' = 0

but then using the D and then Q operators i run into problems when trying to find the particleur integral of

y'' + ay' = -by^n

due to the A term in equation (1), which i cannot use the First Shift Theorem on...

Any ideas? or have i gone wrong somewhere?

NoMoreExams, please, did you even look at the problem? x'' +ax' + bx^n = 0 is non-linear because of that 'x^n' term. None of your suggestions help here.
Pete69, if you cannot solve a simple linear equation like x"+ ax= 0 you certainly cannot expect to solve a difficult non-linear problem like this! Looks like the blind leading the blind here. Pete69, where did you get this problem? Is it for a course?

if you look at my other topic (a few below this one in this section, about free fall under gravity) you will see where it came from... iv only just come across linear constant coefficent second order DEs (homogenous and non-homogenous) so now know how to solve the x'' + ax = 0 part... but my course doesn't cover non-linear DEs, until next yr, or at all.. so have no clue how to solve them, but i was jst interested in furthering my knowledge..

## 1. What is a non-linear second order differential equation?

A non-linear second order differential equation is an equation that involves the second derivative of a function as well as the function itself, and the function is raised to a power or multiplied by a coefficient. The equation does not follow a linear pattern and cannot be solved by simple algebraic methods.

## 2. How do I solve a non-linear second order differential equation?

There is no one method for solving all non-linear second order differential equations. Some common techniques include substitution, integration, and using series solutions. The specific method used will depend on the structure and complexity of the equation.

## 3. Can a non-linear second order differential equation have multiple solutions?

Yes, a non-linear second order differential equation can have multiple solutions. This is because there are often multiple ways to rearrange and simplify the equation, resulting in different solutions. It is important to check the validity of each solution by plugging it back into the original equation.

## 4. Is it possible to solve a non-linear second order differential equation analytically?

In some cases, it is possible to solve a non-linear second order differential equation analytically, meaning that the solution can be written in a closed form expression. However, this is not always possible and numerical methods may need to be used to approximate the solution.

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

Non-linear second order differential equations are commonly used in physics, engineering, and other fields to model natural phenomena such as motion, heat transfer, and chemical reactions. They can also be used in economics and finance to model complex systems and behaviors.

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