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I have the following second order linear ordinary differential equation with variable coefficients:

$ c \frac{d^2 \overline{\varphi}}{d E^2} + aE^k \frac{d \overline{\varphi}}{d E} + (akE^{k-1} +s)\overline{\varphi} = \varphi(E,0)$, where $\overline{\varphi} = \overline{\varphi}(E,s) = \int_0^\infty \varphi(E,x)e^{-sx}dx$.

Before I actually show how I tried to solve this, it is perhaps good if I provide some background. This equation is obtained by taking the Laplace transform in the x variable of the following second order partial differential equation:

$c \cdot \frac{\partial^2 \varphi(E,x)}{\partial E^2} + aE^k \cdot \frac{\partial \varphi(E,x)}{\partial E} + akE^{k-1} \cdot \varphi + \frac{\partial \varphi}{\partial x} = 0 $ where $E \in (\infty,+\infty) , x\in(0,\infty)$ .

Coming back to the first equation, in order to solve this I have attempted to reduce this to an equation with constant coefficients, by following this document: http://www.mecheng.iisc.ernet.in/~sonti/ME261_variable_coeff_2nd_order.pdf.

However, when calculating whether the equation is exact, via the $P''-Q'+R$ formula, the result is the $s $ constant. Thus, I have concluded that the equation cannot be reduced to an equation with constant coefficients.

I have also tried using the variation of parameters on the equation. This was mostly by following the method given here: http://tutorial.math.lamar.edu/Classes/DE/VariationofParameters.aspx

However, that requires knowledge of the fundamental solutions of the equation. The fundamental solutions are obtained by solving the following equation:

$ c \frac{d^2 \overline{\varphi}}{d E^2} + aE^k \frac{d \overline{\varphi}}{d E} + (akE^{k-1} +s)\overline{\varphi} = 0$

I attempted thereafter to solve the following polynomial:

$cr^2+aE^kr+(akE^{k-1}+s)=0$

The constants in the equation are: $c = 0.1/2$, $a = e^{148}$ and $k = -1.4$. When I plotted the delta from the well known abc formula it always turned out positive. So I have concluded that the roots of the equation are real and thus the fundamental solutions are

$\overline{\varphi}_1 = e^{Er_1(s)}$ and $\overline{\varphi}_2 = e^{Er_2(s)}$, with $r_1$ and $r_2$ the roots of the characteristic polynomial.

Thus, the variation of parameters method states that the solution to this equation is:

$\overline{\varphi}(E,s) = -\overline{\varphi_1} \int_0^\infty \frac{\overline{\varphi_2}\varphi(E,0)}{W(\overline{\varphi_1},\overline{\varphi_2})} + \overline{\varphi_2} \int_0^\infty \frac{\overline{\varphi_1}\varphi(E,0)}{W(\overline{\varphi_1},\overline{\varphi_2})}$

Using this the solution to the original PDE can be obtained via:

$\varphi(E,x) = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \overline{\varphi}(E,s)e^{sx}ds$

So, my questions are:

  1. Does the first equation in the post have an analytical solution?
  2. Is my treatment correct?
  3. Do you have suggestions for improvements/ alternative methods?

Thank you in advance

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Hint:

For $c\dfrac{d^2\overline\varphi}{dE^2}+aE^k\dfrac{d\overline\varphi}{dE}+(akE^{k-1}+s)\overline\varphi=0$ ,

Let $\overline\varphi=e^{mE}\overline U$ ,

Then $\dfrac{d\overline\varphi}{dE}=e^{mE}\dfrac{d\overline U}{dE}+me^{mE}\overline U$

$\dfrac{d^2\overline\varphi}{dE^2}=e^{mE}\dfrac{d^2\overline U}{dE^2}+me^{mE}\dfrac{d\overline U}{dE}+me^{mE}\dfrac{d\overline U}{dE}+m^2e^{mE}\overline U=e^{mE}\dfrac{d^2\overline U}{dE^2}+2me^{mE}\dfrac{d\overline U}{dE}+m^2e^{mE}\overline U$

$\therefore c\left(e^{mE}\dfrac{d^2\overline U}{dE^2}+2me^{mE}\dfrac{d\overline U}{dE}+m^2e^{mE}\overline U\right)+aE^k\left(e^{mE}\dfrac{d\overline U}{dE}+me^{mE}\overline U\right)+(akE^{k-1}+s)e^{mE}\overline U=0$

$c\left(\dfrac{d^2\overline U}{dE^2}+2m\dfrac{d\overline U}{dE}+m^2\overline U\right)+aE^k\left(\dfrac{d\overline U}{dE}+m\overline U\right)+(akE^{k-1}+s)\overline U=0$

$c\dfrac{d^2\overline U}{dE^2}+(aE^k+2cm)\dfrac{d\overline U}{dE}+(amE^k+akE^{k-1}+cm^2+s)\overline U=0$

Take $cm^2+s=0$ , i.e. $m=\pm i\sqrt{\dfrac{s}{c}}$ , the ODE becomes

$c\dfrac{d^2\overline U}{dE^2}+(aE^k\pm2i\sqrt{cs})\dfrac{d\overline U}{dE}+\left(\pm ia\sqrt{\dfrac{s}{c}}E^k+akE^{k-1}\right)\overline U=0$

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