28.3.8 problem 9

Internal problem ID [7182]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter VII, Solutions in series. Examples XIV. page 177
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:25:07 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 36

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

\[ y = c_1 \left (1-\frac {2}{3} x +\frac {1}{6} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-2+8 x -12 x^{2}+8 x^{3}-2 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 40

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

\[ y(x)\to c_1 \left (x^2+\frac {1}{x^2}-4 x-\frac {4}{x}+6\right )+c_2 \left (\frac {x^2}{6}-\frac {2 x}{3}+1\right ) \]

✓ Sympy. Time used: 0.333 (sec). Leaf size: 34

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

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{9450} + \frac {x^{4}}{540} + \frac {x^{3}}{45} + \frac {x^{2}}{6} + \frac {2 x}{3} + 1\right ) + O\left (x^{6}\right ) \]