76.47.1 problem Ex. 1

Internal problem ID [20261]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. Exact differential equations, and equations of particular forms. Integration in series. problems at page 102
Problem number : Ex. 1
Date solved : Thursday, October 02, 2025 at 05:38:03 PM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (-x^{2}+x \right ) y^{\prime \prime }+4 y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 38

Order:=6; 
ode:=(-x^2+x)*diff(diff(y(x),x),x)+4*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{2} x +\frac {1}{10} x^{2}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (12-60 x +120 x^{2}-120 x^{3}+60 x^{4}-12 x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{3}} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 42

ode=(x-x^2)*D[y[x],{x,2}]+4*D[y[x],x]+2*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {x^2}{10}-\frac {x}{2}+1\right )+c_1 \left (\frac {1}{x^3}-\frac {5}{x^2}+5 x+\frac {10}{x}-10\right ) \]

✓ Sympy. Time used: 0.318 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-x**2 + x)*Derivative(y(x), (x, 2)) + 2*y(x) + 4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \left (\frac {x^{5}}{25200} + \frac {x^{4}}{1260} + \frac {x^{3}}{90} + \frac {x^{2}}{10} + \frac {x}{2} + 1\right ) + O\left (x^{6}\right ) \]