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1st order linear differential equation

  • Thread starter musicmar
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  • #1
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Homework Statement


I'm trying to study for a quiz tomorrow by doing some practice problems. If someone could help me with the process of solving a 1st order linear diff. eq., that would be great.

(x+1)(dy/dx) + (x+2)y = 2xe-x



Homework Equations





The Attempt at a Solution



dy/dx + [(x+2)/(x+1)]y = 2xe-x/(x+1)

integrating factor: e∫(x+2)/(x+1)= exlx+1l

This is where I get confused. I should be able to use the product rule here:

(y(exlx+1l)'

so that I will be able to take the integral of (above) and {2xe-x/(x+1)]*[exlx+1l].

Once I take the integrals, then I can solve for c(not in this problem, though) and try to solve for y explicitly. Some help with the middle steps would be greatly appreciated.
 

Answers and Replies

  • #2
hunt_mat
Homework Helper
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First divide throughout to obtain an equation of the form:
[tex]
\frac{dy}{dx}+P(x)y=Q(x)
[/tex]
Then multiply through by the integrating factor and the LHS will be a total derivative, in your case it should be:
[tex]
\left( e^{x}(1+x)y\right) '=2x
[/tex]
 
  • #3
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Where did the 2x come from?
 
  • #4
HallsofIvy
Science Advisor
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The point was that the formula you used for the integrating factor requires that the coefficient of the derivative be 1. Here it is x+ 1 so you need to divide the entire equation by x+1:
[tex]\frac{dy}{dx}+ \frac{x+2}{x+1}y= \frac{2x}{x+1}e^{-x}[/tex].
(The first equality is from the product rule, the second from just multiplying the left side of the differential equation by u.)

Now, you are looking for a function, u(x), so that multiplying by it will make that left side a single derivative:
[tex]\frac{d(u(x)y)}{dx}= u(x)\frac{dy}{dx}+ \frac{du}{dx}y= u\frac{dy}{dx}+ \frac{x+2}{x+1}u y[/itex]
That is, we must have
[tex]\frac{du}{dx}= \frac{x+2}{x+1}u[/tex]
or
[tex]\frac{du}{u}= \frac{x+2}{x+1}dx= (1+ \frac{1}{x+2})dx[/tex]

Integrating both sides, [itex]ln(u)= x+ ln(x+2)[/itex] so that
[tex]u(x)= e^{x+ ln(x+2)}= (x+ 2)e^x[/tex]

What do you get when you multiply both sides of your equation by that?
 
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  • #5
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dy/dx + [(x+2)/(x+1)]y = 2xe-x/(x+1)

y'(x+2)ex+[(x+2)2/(x+1)]y=2x

Ok, now I see where the 2x comes from. Do I have the rest right?

If so, then by the product rule I should have:

(ex(x+2)y)'=2x

Taking the integral of both sides:

ex(x+2)y=x2+c

y=(x2+c)/(ex(x+2))


Yes?
 

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