Solve y'''=y'': Using Power Series to Solve Non-Linear Diff Eq

[lkj-17]

How to use power series to solve this non-linear differential equation?

 

[HallsofIvy]

First, because this is a third order equation, its general solution will involve 3 undetermined constants. So, assume y(0)= A, y'(0)= B, y"(0)= C.

From y'y"'= y", y"'= y"/y' and so y"'(0)= C/B. Now, differentiating y"'= y"/y' again, y^iv= (y'y"'- y"²)/y'² so y^iv(0)= (B(C/B)- C^2)/B²= (C- C²)/B².

So far, we have y(x)= y(0)+ y'(0)x+ (1/2)y"(0)x²+ (1/3!)y"'(0)x³+ (1/4!)y^iv(0)x⁴+ ...= A+ Bx+ (1/2)Cx²+ (1/6)(C/B)x³+ (1/24){(C- C²)/B²}x^4+ ...

Continue like that to get higher terms.

 

[Matthew Rodman]

Re-arrange your equation as

$$y^{\prime \prime \prime} = \frac{y^{\prime \prime}}{y^{\prime}}$$

Now integrate with respect to x to get

$$y^{\prime \prime} = \kappa + \ln{y^{\prime}}$$

where $$\kappa$$ is a constant of integration. Now re-arrange and integrate to get

$$\int{\frac{d y^{\prime}}{\kappa + \ln{y^{\prime}}} = x + \epsilon$$

where $$\epsilon$$ is another constant.

I checked the Integrator (Wolfram site), and it gave the integral as:

$$\int{\frac{dw}{a + \ln{w}}} = e^{-a}Ei(a + \ln{w})$$

where $$Ei$$ is the Exponential Integral.

Hence we can apply this to our integral to get

$$e^{- \kappa} Ei(\kappa + \ln{y^{\prime}) = x + \epsilon$$

which we can re-arrange as

$$y^{\prime} = exp(Ei^{-1}(e^{\kappa}(x + \epsilon)) - \kappa)$$

We can tidy this up a bit by making

$$e^{\kappa} = \alpha$$

and

$$\epsilon \alpha = \beta$$

and then integrate to get

$$y(x) = \frac{1}{\beta}\int{exp(Ei^{-1}(\alpha x + \beta)) dx}$$

and then you'll have to try some numerical techniques to obtain values for y(x) (I have to idea how to express the inverse of "Ei".)

 

[Matthew Rodman]

Sorry that should read:

$$y(x) = \frac{1}{\alpha}\int{exp(Ei^{-1}(\alpha x + \beta)) dx}$$