Equation of Paths for Second order ODE: d^y/dx^2 - x - 2x^3 = 0

Belief7

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Nov 6, 2018
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Find the equation of the paths of 𝑑 2𝑦 𝑑π‘₯ 2 βˆ’ π‘₯ + 2π‘₯ 3 = 0 sketch the paths in the plane. Locate their critical points and determine its nature.

Can someone please provide a solution with an easy to understand explanation?
 

Belief7

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Hi,

The question is as follows:
Find the equation of the paths of 𝑑²y/𝑑π‘₯Β² βˆ’ π‘₯ + 2π‘₯Β³ = 0. Sketch the paths in the plane.
Locate their critical points and determine its nature.

I have calculated the system of LDE to be:
dy/dx = t + 0*y
dt/dx d= x - 2*xΒ³

I am stuck in the same step.
Please help me proceed further,.
 

HallsofIvy

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Your differential equation is \(\displaystyle \frac{d^2y}{dx^2}- x- 2x^3= 0\) and you are asked to "sketch the paths in the xy-plane"?

I would just write the equation as \(\displaystyle \frac{d^2y}{dx^2}= 2x^3+ x\) and integrate twice:
\(\displaystyle \frac{dy}{dx}= \frac{1}{2}x^4+ \frac{1}{2}x^2+ C\)
\(\displaystyle y(x)= \frac{1}{10}x^5+\frac{1}{6}x^3+ Cx+ D\)


Each choice of C and D gives a different curve.
 

Belief7

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Nov 6, 2018
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Your differential equation is \(\displaystyle \frac{d^2y}{dx^2}- x- 2x^3= 0\) and you are asked to "sketch the paths in the xy-plane"?

I would just write the equation as \(\displaystyle \frac{d^2y}{dx^2}= 2x^3+ x\) and integrate twice:
\(\displaystyle \frac{dy}{dx}= \frac{1}{2}x^4+ \frac{1}{2}x^2+ C\)
\(\displaystyle y(x)= \frac{1}{10}x^5+\frac{1}{6}x^3+ Cx+ D\)


Each choice of C and D gives a different curve.
thank you! that helped! Although, I was wondering whether we could we first formulate a system of linear DE from the given second-order DE rather than just separating the variables and integrating. I would like to know how you would do that.
 

HallsofIvy

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The standard method would be to introduce the new variable, \(\displaystyle u= \frac{dy}{dx}\). Then \(\displaystyle \frac{d^2y}{dx^2}= \frac{du}{dx}= 2x^3+ x\).

Now you have the system of equations
\(\displaystyle \frac{dy}{dx}= u\)
\(\displaystyle \frac{du}{dx}= 2x^3+ x\).

Since the second equation does not involve y those equations are "uncoupled" and the simplest way to handle this is to integrate the second equation directly.
 
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