84.30.4 problem 18.14

Internal problem ID [22293]
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 : 18.14
Date solved : Thursday, October 02, 2025 at 08:37:11 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*} -1 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 64

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=-1);
 

\[ y = c_1 \left (x +1\right )^{2} \left (1+\left (x +1\right )+\frac {5}{8} \left (x +1\right )^{2}+\frac {19}{40} \left (x +1\right )^{3}+\frac {389}{960} \left (x +1\right )^{4}+\frac {11747}{33600} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right )+c_2 \left (\ln \left (x +1\right ) \left (3 \left (x +1\right )^{2}+3 \left (x +1\right )^{3}+\frac {15}{8} \left (x +1\right )^{4}+\frac {57}{40} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right )+\left (-2+2 \left (x +1\right )+2 \left (x +1\right )^{2}-\left (x +1\right )^{3}-\frac {21}{32} \left (x +1\right )^{4}-\frac {449}{2400} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 108

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,-1,5}]
 

\[ y(x)\to c_2 \left (\frac {389}{960} (x+1)^6+\frac {19}{40} (x+1)^5+\frac {5}{8} (x+1)^4+(x+1)^3+(x+1)^2\right )+c_1 \left (\frac {1}{64} \left (91 (x+1)^4+144 (x+1)^3+48 (x+1)^2-64 (x+1)+64\right )-\frac {3}{16} (x+1)^2 \left (5 (x+1)^2+8 (x+1)+8\right ) \log (x+1)\right ) \]

✓ Sympy. Time used: 0.427 (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=-1,n=6)
 

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