84.30.3 problem 18.13

Internal problem ID [22292]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 18. Linear differential equations with variable coefficients. Supplementary problems
Problem number : 18.13
Date solved : Thursday, October 02, 2025 at 08:37:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 38

Order:=6; 
ode:=(1+x)*diff(diff(y(x),x),x)+1/x*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-\frac {1}{9} x^{3}+\frac {1}{24} x^{4}-\frac {1}{50} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\frac {2}{27} x^{3}-\frac {11}{144} x^{4}+\frac {33}{1000} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 82

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

\[ y(x)\to c_1 \left (-\frac {x^5}{50}+\frac {x^4}{24}-\frac {x^3}{9}+1\right )+c_2 \left (\frac {33 x^5}{1000}-\frac {11 x^4}{144}+\frac {2 x^3}{27}+\left (-\frac {x^5}{50}+\frac {x^4}{24}-\frac {x^3}{9}+1\right ) \log (x)+x\right ) \]

✓ Sympy. Time used: 0.342 (sec). Leaf size: 7

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

\[ y{\left (x \right )} = C_{1} + O\left (x^{6}\right ) \]