20.25.15 problem 16

Internal problem ID [4020]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.4. page 758
Problem number : 16
Date solved : Monday, January 27, 2025 at 08:06:22 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1-x \right ) y^{\prime }-\left (5+x \right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 233

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*(1-x)*diff(y(x),x)-(5+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{-\sqrt {5}} \left (1+\frac {\sqrt {5}-1}{-1+2 \sqrt {5}} x +\frac {\sqrt {5}-2}{-4+8 \sqrt {5}} x^{2}+\frac {\left (\sqrt {5}-2\right ) \left (\sqrt {5}-3\right )}{276-96 \sqrt {5}} x^{3}+\frac {\left (\sqrt {5}-3\right ) \left (\sqrt {5}-4\right )}{2208-768 \sqrt {5}} x^{4}+\frac {\left (\sqrt {5}-3\right ) \left (\sqrt {5}-4\right ) \left (-5+\sqrt {5}\right )}{41280 \sqrt {5}-93600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{\sqrt {5}} \left (1+\frac {\sqrt {5}+1}{1+2 \sqrt {5}} x +\frac {\sqrt {5}+2}{8 \sqrt {5}+4} x^{2}+\frac {\left (\sqrt {5}+2\right ) \left (3+\sqrt {5}\right )}{276+96 \sqrt {5}} x^{3}+\frac {\left (3+\sqrt {5}\right ) \left (\sqrt {5}+4\right )}{2208+768 \sqrt {5}} x^{4}+\frac {\left (3+\sqrt {5}\right ) \left (\sqrt {5}+4\right ) \left (5+\sqrt {5}\right )}{41280 \sqrt {5}+93600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 1093

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*(1-x)*D[y[x],x]-(5+x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \left (\frac {\left (-5-\sqrt {5}\right ) \left (-4-\sqrt {5}\right ) \left (-3-\sqrt {5}\right ) \left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^5}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-2+\sqrt {5}+\left (2+\sqrt {5}\right ) \left (3+\sqrt {5}\right )\right ) \left (-1+\sqrt {5}+\left (3+\sqrt {5}\right ) \left (4+\sqrt {5}\right )\right ) \left (\sqrt {5}+\left (4+\sqrt {5}\right ) \left (5+\sqrt {5}\right )\right )}-\frac {\left (-4-\sqrt {5}\right ) \left (-3-\sqrt {5}\right ) \left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^4}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-2+\sqrt {5}+\left (2+\sqrt {5}\right ) \left (3+\sqrt {5}\right )\right ) \left (-1+\sqrt {5}+\left (3+\sqrt {5}\right ) \left (4+\sqrt {5}\right )\right )}+\frac {\left (-3-\sqrt {5}\right ) \left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^3}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right ) \left (-2+\sqrt {5}+\left (2+\sqrt {5}\right ) \left (3+\sqrt {5}\right )\right )}-\frac {\left (-2-\sqrt {5}\right ) \left (1+\sqrt {5}\right ) x^2}{\left (-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )\right ) \left (-3+\sqrt {5}+\left (1+\sqrt {5}\right ) \left (2+\sqrt {5}\right )\right )}+\frac {\left (1+\sqrt {5}\right ) x}{-4+\sqrt {5}+\sqrt {5} \left (1+\sqrt {5}\right )}+1\right ) c_1 x^{\sqrt {5}}+\left (\frac {\left (1-\sqrt {5}\right ) \left (-5+\sqrt {5}\right ) \left (-4+\sqrt {5}\right ) \left (-3+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^5}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-2-\sqrt {5}+\left (2-\sqrt {5}\right ) \left (3-\sqrt {5}\right )\right ) \left (-1-\sqrt {5}+\left (3-\sqrt {5}\right ) \left (4-\sqrt {5}\right )\right ) \left (-\sqrt {5}+\left (4-\sqrt {5}\right ) \left (5-\sqrt {5}\right )\right )}-\frac {\left (1-\sqrt {5}\right ) \left (-4+\sqrt {5}\right ) \left (-3+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^4}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-2-\sqrt {5}+\left (2-\sqrt {5}\right ) \left (3-\sqrt {5}\right )\right ) \left (-1-\sqrt {5}+\left (3-\sqrt {5}\right ) \left (4-\sqrt {5}\right )\right )}+\frac {\left (1-\sqrt {5}\right ) \left (-3+\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^3}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right ) \left (-2-\sqrt {5}+\left (2-\sqrt {5}\right ) \left (3-\sqrt {5}\right )\right )}-\frac {\left (1-\sqrt {5}\right ) \left (-2+\sqrt {5}\right ) x^2}{\left (-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )\right ) \left (-3-\sqrt {5}+\left (1-\sqrt {5}\right ) \left (2-\sqrt {5}\right )\right )}+\frac {\left (1-\sqrt {5}\right ) x}{-4-\sqrt {5}-\sqrt {5} \left (1-\sqrt {5}\right )}+1\right ) c_2 x^{-\sqrt {5}} \]