82.1.3 problem 23-8

Internal problem ID [21745]
Book : The Differential Equations Problem Solver. VOL. II. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 23. Power series. Page 695
Problem number : 23-8
Date solved : Thursday, October 02, 2025 at 08:01:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+7 x \left (1+x \right ) y^{\prime }-3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

Order:=6; 
ode:=2*x^2*diff(diff(y(x),x),x)+7*x*(1+x)*diff(y(x),x)-3*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{7}/{2}} \left (1-\frac {7}{18} x +\frac {49}{264} x^{2}-\frac {1715}{20592} x^{3}+\frac {16807}{494208} x^{4}-\frac {117649}{9335040} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {21}{5} x +\frac {49}{5} x^{2}-\frac {343}{15} x^{3}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

ode=2*x^2*D[y[x],{x,2}]+7*x*(x+1)*D[y[x],x]-3*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_2 \left (-\frac {343 x^3}{15}+\frac {49 x^2}{5}-\frac {21 x}{5}+1\right )}{x^3}+c_1 \sqrt {x} \left (-\frac {117649 x^5}{9335040}+\frac {16807 x^4}{494208}-\frac {1715 x^3}{20592}+\frac {49 x^2}{264}-\frac {7 x}{18}+1\right ) \]

✓ Sympy. Time used: 0.334 (sec). Leaf size: 65

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*Derivative(y(x), (x, 2)) + 7*x*(x + 1)*Derivative(y(x), x) - 3*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt {x} \left (\frac {16807 x^{4}}{494208} - \frac {1715 x^{3}}{20592} + \frac {49 x^{2}}{264} - \frac {7 x}{18} + 1\right ) + \frac {C_{1} \left (- \frac {343 x^{3}}{15} + \frac {49 x^{2}}{5} - \frac {21 x}{5} + 1\right )}{x^{3}} + O\left (x^{6}\right ) \]