Help solving non-linear second order DE

[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

 

[NoMoreExams]

Don't you solve the associated quadratic?

 

[Pete69]

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

 

[NoMoreExams]

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?

 

[Pete69]

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

cheers

 

[Pete69]

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

 

[NoMoreExams]

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

 

[NoMoreExams]

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/

 

[Pete69]

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..

 

[Pete69]

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?

 

[HallsofIvy]

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?

 

[Pete69]

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..