A relatively simple ODE problem

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In summary: You should get y = 1/(x^2(x^3 - 2))In summary, the conversation discusses a student struggling with a differential equations problem and seeking help from others. The problem involves a non-separable and non-linear equation, requiring the use of an integrating factor and a change of variable. The student works through multiple steps and equations to arrive at a solution, but makes a few mistakes along the way.
  • #1
bennyska
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So many years ago, i took elementary differential equations. unfortunately, I've forgotten too much of it, and now I'm in a not so elementary class. I'm confident i'll catch up to where i was, but at the moment I'm stuck on a relatively simple problem.


i'm having trouble separating them (not sure if this is separable). maybe if someone could give me a pointer to get started, i would be very appreciative. thanks!

Homework Statement




x3 + 3y -xy' = 0.

Homework Equations





The Attempt at a Solution


stuck
 
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  • #2
It doesn't look separable to me. Try finding an integrating factor. Does that ring a bell?
 
  • #3
okay, so I'm not getting the correct answer, but here's the work I've done so far:
x3 + 3y -xy' = 0
3y - xy' = -x3
x(3y/x) - y' = -x3
-y' + 3/x*y = -x2

let mu(x) = eintegral(-3/x) = e-3 ln x = 1/x3

d/dx[1/x3*y] = 1/x3* x2
1/x3*y = x-1
y=x2

is this the right path?

editing:
i saw i didn't integrate the left hand side at x-1. afterwards, it seemed to work.
y = x3ln(x)
 
Last edited:
  • #4
Almost. You forgot the constant of integration, which will result in another term, cx3, which is the solution to the homogeneous equation.
 
  • #5
cool, thanks so much.
 
  • #6
it's slowly coming back to me. okay, linear equations, put them in the proper form, use integrating factor, got it.
now I'm kind of stuck again with a problem from the same set.

xy + y2 - x2y'=0.

so this isn't separable, and it's not linear, and I'm kind of stuck. any help, or advice where to start looking? gracias.
 
  • #7
If you divide both sides by x^2, you've got y/x+(y/x)^2-y'=0. Suggests that a change of variable v=y/x might help.
 
  • #8
how's this:

[tex]\frac{y}{x}[/tex] + ([tex]\frac{y}{x}[/tex])2 - y' = 0
let v= [tex]\frac{y}{x}[/tex], with [tex]\frac{dy}{dx}[/tex] = v + x[tex]\frac{dv}{dx}[/tex]

v + v2 - v - x[tex]\frac{dv}{dx}[/tex] = 0
v2 - x[tex]\frac{dv}{dx}[/tex] = 0

[tex]\frac{1}{v2}[/tex] dv = [tex]\frac{1}{x}[/tex]dx

integrate, get

v = [tex]\frac{-1}{ln x}[/tex]

substitute and solve for y

y = x ([tex]\frac{-1}{ln x}[/tex]

and do i just throw a constant on at the end?
 
  • #9
It's pretty good. But you don't just "throw a constant on at the end". You put in a constant at the point you integrate then carry it through.
 
  • #10
yeah, i kind of figured that, it just seems that that's how it usually works out, and i can't remember to put them in when they come up.
so I'm having another hard time working out a different problem.

(for future reference, should i start a new thread if i have a different problem?)

so it's a bernoulli equation
4xy2 + y' = 5x4y2
y' = 5x4y2 - 4xy2
y' = y2( 5x4 - 4x)

let n = 2, p(x) = 0, g(x) = 5x4 - 4x, v = y-1
divide both sides by y2
y-2y' + 0y = 5x4 - 4x
substitute v= y-1

-dv/dx + 0v = 5x4 - 4x
let mu(x) = eint. 0 dx = ec0 = c1
-c1 dv/dx + 1*0v = c1(5x4 - 4x)
d/dx c1*v = c1(5x4 - 4x)
integrate both sides
c1*v=-c1*x2(x3 + 2)
v = -x2(x3 + 2)
replace v
y-1 = -x2(x3 + 2)
y = 1/ (-x2(x3 + 2))
however, this doesn't seem to work. I'm guessing maybe letting p(x) = 0 might be a problem? i don't see why, it's continuous, but that's the only thing i can think of.

btw, thanks for your help. it's coming back to me. i was able to solve 2 other bernoulli equations and got stuck on this one.
 
  • #11
bennyska said:
yeah, i kind of figured that, it just seems that that's how it usually works out, and i can't remember to put them in when they come up.
so I'm having another hard time working out a different problem.

(for future reference, should i start a new thread if i have a different problem?)

so it's a bernoulli equation
4xy2 + y' = 5x4y2
y' = 5x4y2 - 4xy2
y' = y2( 5x4 - 4x)
You're making things hard for yourself. You should be able to see at this point that the equation is separable.
 
  • #12
bennyska said:
let n = 2, p(x) = 0, g(x) = 5x4 - 4x, v = y-1
divide both sides by y2
y-2y' + 0y = 5x4 - 4x
substitute v= y-1

-dv/dx + 0v = 5x4 - 4x
let mu(x) = eint. 0 dx = ec0 = c1
-c1 dv/dx + 1*0v = c1(5x4 - 4x)
d/dx c1*v = c1(5x4 - 4x)
integrate both sides
c1*v=-c1*x2(x3 + 2)
v = -x2(x3 + 2)
replace v
y-1 = -x2(x3 + 2)
y = 1/ (-x2(x3 + 2))
however, this doesn't seem to work. I'm guessing maybe letting p(x) = 0 might be a problem? i don't see why, it's continuous, but that's the only thing i can think of.

btw, thanks for your help. it's coming back to me. i was able to solve 2 other bernoulli equations and got stuck on this one.
You dropped a sign or two, and again, you forgot the constant of integration.
 

FAQ: A relatively simple ODE problem

What is an ODE?

An ODE (Ordinary Differential Equation) is a mathematical equation that describes the relationship between a function and its derivatives. It involves one independent variable and one or more dependent variables.

What makes a problem a "relatively simple" ODE problem?

A "relatively simple" ODE problem typically refers to an ODE that can be solved analytically using standard methods, such as separation of variables, substitution, or integration. It does not involve complex or higher-order derivatives.

Why are ODEs important in science?

ODEs are used to model and describe many physical phenomena in science and engineering. They are particularly useful in studying systems that change over time, such as population growth, chemical reactions, and motion of objects.

How do you solve an ODE problem?

There are various methods for solving ODE problems, including analytical and numerical methods. Analytical methods involve finding an exact solution using mathematical techniques, while numerical methods use algorithms to approximate the solution. The method used depends on the type and complexity of the ODE.

Can ODE problems have real-world applications?

Yes, ODE problems have a wide range of real-world applications in fields such as physics, biology, economics, and engineering. They are used to model and predict the behavior of systems and can provide valuable insights and solutions to real-world problems.

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