43.17.9 problem 2(c)

Internal problem ID [9002]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 154
Problem number : 2(c)
Date solved : Tuesday, September 30, 2025 at 06:01:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 51

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

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {1}{2} x^{2}-\frac {1}{15} x^{3}+\frac {1}{72} x^{4}+\frac {137}{1950} x^{5}+\frac {307}{36720} x^{6}-\frac {7169}{3439800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \,x^{2} \left (1-\frac {1}{30} x^{2}-\frac {8}{57} x^{3}+\frac {1}{2760} x^{4}+\frac {64}{12825} x^{5}+\frac {147181}{9753840} x^{6}-\frac {4037}{72268875} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 106

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

\[ y(x)\to c_1 \left (-\frac {4037 x^7}{72268875}+\frac {147181 x^6}{9753840}+\frac {64 x^5}{12825}+\frac {x^4}{2760}-\frac {8 x^3}{57}-\frac {x^2}{30}+1\right ) x^2+c_2 \left (-\frac {7169 x^7}{3439800}+\frac {307 x^6}{36720}+\frac {137 x^5}{1950}+\frac {x^4}{72}-\frac {x^3}{15}-\frac {x^2}{2}+1\right ) \sqrt [4]{x} \]

✓ Sympy. Time used: 0.414 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*Derivative(y(x), (x, 2)) + (x**2 + 2)*y(x) + (4*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)
 

\[ y{\left (x \right )} = C_{2} x^{2} \left (\frac {64 x^{5}}{12825} + \frac {x^{4}}{2760} - \frac {8 x^{3}}{57} - \frac {x^{2}}{30} + 1\right ) + C_{1} \sqrt [4]{x} \left (\frac {307 x^{6}}{36720} + \frac {137 x^{5}}{1950} + \frac {x^{4}}{72} - \frac {x^{3}}{15} - \frac {x^{2}}{2} + 1\right ) + O\left (x^{8}\right ) \]