how to solve a 3rd order differential equation with non-constant coefficients

I am trying to solve this third order differential equation with non-constant coefficients $$(x^3)y'''+6 x^2 y''+[6+(1+a-bx^2)x^2]x y'+[1+3 a-5 b x^2]x^2 y=0$$ where $$a$$ and $$b$$ are constants and $$y$$ is a function of $$x$$ only.

The problem originated form a Micropolar fluid flow problem involving longitudinal and torsional oscillations. The original differential equation was 4th order and a combination of the Laplace transform, a change in variables as well as multiple integral transforms were used to bring the form given below.

I have tried the standard methods for solving differential equations with variable coefficients, as well as the Frobenius power series method, using x =0 as a regular singular point, however it became too complicated for a pattern and hence solutions to be formed.

I would be grateful if anyone had any further ideas on how to solve this differential equation.

• Is there some reason you did not add the 6 to the $1+a$? – John Wayland Bales Feb 4 at 17:43
• @JohnWaylandBales ty I will correct the issue now, that term is multiplied by x squared – user447631 Feb 4 at 17:45
• It looked promising, but didn't quite work. I'll post in case it offers inspiration. Can re-write as $(x^3 y'')' + 3(x^2y')' +((ax^3+bx^5)y)'+x^3y' + x^2 y=0$. If only it was $3x^2 y$ at the end. – user121049 Feb 4 at 18:15
• @user121049 thanks so much for this- to get to this differential equation I had about 50 pages of pen to paper working, I will redo and make sure I do not have any errors that would allow for the 3 x-squared y- much appreciated! – user447631 Feb 4 at 19:14

The indicial roots are $$0$$, $$-1$$, $$-2$$. There is a series solution of the form $$\sum_{k=0}^\infty c_{2k} x^{2k}$$ with $$c_0 = 1$$, $$c_2 = -a/8 - 1/24$$, and

$$-(n+5) b c_n + ((a+1)n + 5a + 3) c_{n+2} + + (n+4)(n+5)(n+6) c_{n+4} = 0$$

and a series solution of the form $$\sum_{k=0}^\infty c_{2k-1} x^{2k-1}$$ with $$c_{-1}=1$$, $$c_1 = -a/3$$, and this same recurrence. A third fundamental solution involves $$x^n$$ for even $$n \ge -2$$ and $$x^n \ln(x)$$ for even $$n \ge 0$$.

• thank you for your help! I got to the indicial roots as well but the pattern became too hard to spot for me when I listed out the terms odd and evens . I will go through this again, much appreciated! – user447631 Feb 4 at 19:15

$$(x^3)y'''+6 x^2 y''+[6+(1+a-bx^2)x^2]x y'+[1+3 a-5 b x^2]x^2 y=0$$ This is not a final answer, only a way to simplify the problem.

One observe that the change of $$x$$ to $$-x$$ doesn't change the equation. This draw us to a change of variable in order to simplify : $$X=x^2\quad;\quad X'=2x \quad;\quad y(x)=u(X)$$ $$y'= 2xu'\quad;\quad y''=4x^2u''+2u' \quad;\quad y'''=8x^3u'''+12xu''$$

The derivatives of $$y$$ are with respect to $$x$$. The derivatives of $$u$$ are with respect to $$X$$.

$$x^3(8x^3u'''+12xu'')+6 x^2 (4x^2u''+2u') +[6+(1+a-bx^2)x^2]x (2xu')+[1+3 a-5 b x^2]x^2 y=0$$

After simplification :

$$8X^2u'''+36Xu'' +[24+2(1+a)X-2bX^2]u'+[1+3 a-5 bX] y=0$$

In the general case, I don't think that a closed form solution exists with the available standard functions.

In the case $$a=0$$ and $$b\neq 0$$ WolframAlpha cannot find a closed form solution.

In the case $$b=0$$ and $$a\neq 0$$ WolframAlpha gives a very complicated closed form solution involving hypergeometric functions and a Meijer-G function.

In the particular case $$a=0$$ and $$b=0$$ WolframAlpha gives the solution on the form $$u=c_1\frac{1}{\sqrt{X}}+c_2\:_1F_2(\frac12,\frac32;2;-\frac{X}{4})+c_3$$(Meijer-G function).

• thank you very much, i will definitely look into this technique today- in the case where both a and b are zero will turn the problem from a non-newtonian fluid to a newtonian one, which i know has been solved so it makes sense you got a solution there. This opened my eyes a lot on how to approach the problem and I appreciate you taking the time to help me with this. – user447631 Feb 5 at 15:34

Hint:

Let $$r=x^2$$ ,

Then $$\dfrac{dy}{dx}=\dfrac{dy}{dr}\dfrac{dr}{dx}=2x\dfrac{dy}{dr}$$

$$\dfrac{d^2y}{dx^2}=\dfrac{d}{dx}\left(2x\dfrac{dy}{dr}\right)=2x\dfrac{d}{dx}\left(\dfrac{dy}{dr}\right)+2\dfrac{dy}{dr}=2x\dfrac{d}{dr}\left(\dfrac{dy}{dr}\right)\dfrac{dr}{dx}+2\dfrac{dy}{dr}=2x\dfrac{d^2y}{dr^2}2x+2\dfrac{dy}{dr}=4x^2\dfrac{d^2y}{dr^2}+2\dfrac{dy}{dr}=4r\dfrac{d^2y}{dr^2}+2\dfrac{dy}{dr}$$

$$\dfrac{d^3y}{dx^3}=\dfrac{d}{dx}\left(4r\dfrac{d^2y}{dr^2}+2\dfrac{dy}{dr}\right)=\dfrac{d}{dr}\left(4r\dfrac{d^2y}{dr^2}+2\dfrac{dy}{dr}\right)\dfrac{dr}{dx}=2x\left(4r\dfrac{d^3y}{dr^3}+6\dfrac{d^2y}{dr^2}\right)$$

$$\therefore2r^2\left(4r\dfrac{d^3y}{dr^3}+6\dfrac{d^2y}{dr^2}\right)+6r\left(4r\dfrac{d^2y}{dr^2}+2\dfrac{dy}{dr}\right)+(6+(1+a-br)r)2r\dfrac{dy}{dr}+(1+3a-5br)ry=0$$

$$8r^2\dfrac{d^3y}{dr^3}+12r\dfrac{d^2y}{dr^2}+24r\dfrac{d^2y}{dr^2}+12\dfrac{dy}{dr}+(12+2r(1+a-br))\dfrac{dy}{dr}+(1+3a-5br)y=0$$

$$8r^2\dfrac{d^3y}{dr^3}+36r\dfrac{d^2y}{dr^2}-(2br^2-(a+1)r-24)\dfrac{dy}{dr}-(5br-3a-1)y=0$$