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VinnyCee
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Homework Statement
Apply the regular perturbation method to solve the following ordinary differential equation
[tex]\left(1\,+\,\epsilon\,y\right)\,\frac{dy}{dx}\,+\,y\,=\,0[/tex]
subject to
[tex]x\,=\,1;\,y\,=\,1[/tex]
Show that the asymptotic solution is of the form
[tex]y\,=\,e^{1\,-\,x}\,+\,\epsilon\,\left[e^{1\,-\,x}\,-\,e^{2\,(1\,-\,x)}\right]\,+\,\dots[/tex]
Homework Equations
http://www.sm.luth.se/~johanb/applmath/chap2en/index.html"
The Attempt at a Solution
First, I get the base case by setting epsilon to zero.
[tex]\frac{dy_0}{dx}\,+\,y_0\,=\,0[/tex]
Am I supposed to use the "subject to" conditions now?
[tex]\frac{dy_0}{dx}\,=\,-y_0\,=\,-1[/tex]
I have no idea if that is what I am supposed to do, I hope so otherwise I am completely lost here.
Now, I assume that a solution is of the form
[tex]y\,=\,y_0\,+\,\epsilon\,y_1\,+\,\epsilon^2\,y_2\,+\,\dots[/tex]
Differentiating that approximation
[tex]\frac{dy}{dx}\,=\,\frac{dy_0}{dx}\,+\,\epsilon\,\frac{dy_1}{dx}\,+\,\epsilon^2\,\frac{dy_2}{dx}\,+\,\dots[/tex]
Expanding the original equation given
[tex]\frac{dy}{dx}\,+\,\epsilon\,y\,\frac{dy}{dx}\,+\,y\,=\,0[/tex]
and substituting the approximations into it
[tex]\left[\frac{dy_0}{dx}\,+\,\epsilon\,\frac{dy_1}{dx}\,+\,\epsilon^2\,\frac{dy_2}{dx}\,+\,\dots\right]\,+\,\left[\epsilon\,y_0\,\frac{dy_0}{dx}\,+\,2\,\epsilon^2\,y_0\,\frac{dy_1}{dx}\,+\,\epsilon^3\,y_1\,\frac{dy_1}{dx}\,+\,\dots\right]\,+\,\left[y_0\,+\,\epsilon\,y_1\,+\,\epsilon^2\,y_2\,+\,\dots\right]\,=\,0[/tex]
Please tell me if the above substitution and expansion are correct.
Now I start matching terms according to their order.
O(1): [tex]\frac{dy_0}{dx}\,=\,-y_0[/tex]
O([itex]\epsilon[/itex]): [tex]\frac{dy_1}{dx}\,=\,-y_0\,\frac{dy_0}{dx}\,-\,y_1[/tex]
O([itex]\epsilon^2[/itex]): [tex]\frac{dy_2}{dx}\,=\,-2\,y_0\,\frac{dy_1}{dx}\,-\,y_2[/tex]
Solving for order one
[tex]\int\,\frac{dy_0}{dx}\,=\,\int\,-y_0[/tex]
[tex]y_0\,=\,K_0[/tex]
OR using the base case that I am unsure about
[tex]y_0\,=\,1[/tex]
But which one is it?
Thanks in advance for your help!
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