20.25.9 problem 10

Internal problem ID [4014]
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 : 10
Date solved : Monday, January 27, 2025 at 08:06:15 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} 4 x^{2} y^{\prime \prime }+3 x y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 44

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+3*x*diff(y(x),x)+x*y(x)=0,y(x),type='series',x=0);
 

\[ y \left (x \right ) = c_{1} x^{{1}/{4}} \left (1-\frac {1}{5} x +\frac {1}{90} x^{2}-\frac {1}{3510} x^{3}+\frac {1}{238680} x^{4}-\frac {1}{25061400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{3} x +\frac {1}{42} x^{2}-\frac {1}{1386} x^{3}+\frac {1}{83160} x^{4}-\frac {1}{7900200} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 85

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+x*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt [4]{x} \left (-\frac {x^5}{25061400}+\frac {x^4}{238680}-\frac {x^3}{3510}+\frac {x^2}{90}-\frac {x}{5}+1\right )+c_2 \left (-\frac {x^5}{7900200}+\frac {x^4}{83160}-\frac {x^3}{1386}+\frac {x^2}{42}-\frac {x}{3}+1\right ) \]