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 : Sunday, March 30, 2025 at 12:45:54 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*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 51

Order:=8; 
ode:=4*x^2*diff(diff(y(x),x),x)+(2*x^4-5*x)*diff(y(x),x)+(3*x^2+2)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

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

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

ode=4*x^2*D[y[x],{x,2}]+(2*x^4-5*x)*D[y[x],x]+(3*x^2+2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

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

✓ Sympy. Time used: 1.059 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + (3*x**2 + 2)*y(x) + (2*x**4 - 5*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {32 x^{5}}{4275} + \frac {3 x^{4}}{920} - \frac {4 x^{3}}{57} - \frac {x^{2}}{10} + 1\right ) + C_{1} \sqrt [4]{x} \left (- \frac {19 x^{6}}{12240} + \frac {137 x^{5}}{1300} + \frac {x^{4}}{8} - \frac {x^{3}}{30} - \frac {3 x^{2}}{2} + 1\right ) + O\left (x^{8}\right ) \]