78.19.11 problem 3 (b)

Internal problem ID [18430]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 5. Power Series Solutions and Special Functions. Section 29. Regular singular Points. Problems at page 227
Problem number : 3 (b)
Date solved : Tuesday, January 28, 2025 at 11:49:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+\left (2 x^{4}-5 x \right ) y^{\prime }+\left (3 x^{2}+2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 43

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)+(2*x^4-5*x)*diff(y(x),x)+(3*x^2+2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_{1} x^{{1}/{4}} \left (1-\frac {3}{2} x^{2}-\frac {1}{30} x^{3}+\frac {1}{8} x^{4}+\frac {137}{1300} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x^{2} \left (1-\frac {1}{10} x^{2}-\frac {4}{57} x^{3}+\frac {3}{920} x^{4}+\frac {32}{4275} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 78

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]+(2*x^4-5*x)*D[y[x],x]+(3*x^2+2)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {32 x^5}{4275}+\frac {3 x^4}{920}-\frac {4 x^3}{57}-\frac {x^2}{10}+1\right ) x^2+c_2 \left (\frac {137 x^5}{1300}+\frac {x^4}{8}-\frac {x^3}{30}-\frac {3 x^2}{2}+1\right ) \sqrt [4]{x} \]