28.4.6 problem 6

Internal problem ID [7195]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XV. page 194
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:25:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 58

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

\[ y = c_1 x \left (1+\frac {1}{2} x +\frac {1}{4} x^{2}+\frac {7}{48} x^{3}+\frac {91}{960} x^{4}+\frac {637}{9600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (\ln \left (x \right ) \left (x +\frac {1}{2} x^{2}+\frac {1}{4} x^{3}+\frac {7}{48} x^{4}+\frac {91}{960} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1-\frac {1}{4} x^{2}-\frac {1}{12} x^{3}-\frac {17}{576} x^{4}-\frac {311}{28800} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 87

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

\[ y(x)\to c_1 \left (\frac {1}{48} x \left (7 x^3+12 x^2+24 x+48\right ) \log (x)+\frac {1}{576} \left (-185 x^4-336 x^3-720 x^2-1152 x+576\right )\right )+c_2 \left (\frac {91 x^5}{960}+\frac {7 x^4}{48}+\frac {x^3}{4}+\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.263 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{2} x + C_{1} + O\left (x^{6}\right ) \]