50.19.11 problem 3(b)

Internal problem ID [8119]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR POINTS. Page 175
Problem number : 3(b)
Date solved : Monday, January 27, 2025 at 03:44:09 PM
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: 0.017 (sec). Leaf size: 51

Order:=8; 
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);
 

\[ y = c_{1} x^{{1}/{4}} \left (1-\frac {3}{2} x^{2}-\frac {1}{30} x^{3}+\frac {1}{8} x^{4}+\frac {137}{1300} x^{5}-\frac {19}{12240} x^{6}-\frac {7169}{764400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{2} \left (1-\frac {1}{10} x^{2}-\frac {4}{57} x^{3}+\frac {3}{920} x^{4}+\frac {32}{4275} x^{5}+\frac {36287}{9753840} x^{6}-\frac {4037}{16059750} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 106

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,"8"-1}]
 

\[ y(x)\to c_1 \left (-\frac {4037 x^7}{16059750}+\frac {36287 x^6}{9753840}+\frac {32 x^5}{4275}+\frac {3 x^4}{920}-\frac {4 x^3}{57}-\frac {x^2}{10}+1\right ) x^2+c_2 \left (-\frac {7169 x^7}{764400}-\frac {19 x^6}{12240}+\frac {137 x^5}{1300}+\frac {x^4}{8}-\frac {x^3}{30}-\frac {3 x^2}{2}+1\right ) \sqrt [4]{x} \]