87.25.14 problem 14

Internal problem ID [23817]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 5. Series solutions of second order linear equations. Exercise at page 232
Problem number : 14
Date solved : Thursday, October 02, 2025 at 09:45:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-2 \left (1+x \right ) y^{\prime }+2 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 3 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 18

Order:=6; 
ode:=x*diff(diff(y(x),x),x)-2*(1+x)*diff(y(x),x)+2*y(x) = 0; 
ic:=[y(3) = 2, D(y)(3) = 0]; 
dsolve([ode,op(ic)],y(x),type='series',x=3);
 

\[ y = 2-\frac {2}{3} \left (x -3\right )^{2}-\frac {14}{27} \left (x -3\right )^{3}-\frac {8}{27} \left (x -3\right )^{4}-\frac {2}{15} \left (x -3\right )^{5}+\operatorname {O}\left (\left (x -3\right )^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 41

ode=x*D[y[x],{x,2}]-2*(x+1)*D[y[x],x]+2*y[x]==0; 
ic={y[3]==2,Derivative[1][y][3] ==0}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,3,5}]
 

\[ y(x)\to -\frac {2}{15} (x-3)^5-\frac {8}{27} (x-3)^4-\frac {14}{27} (x-3)^3-\frac {2}{3} (x-3)^2+2 \]

✓ Sympy. Time used: 0.284 (sec). Leaf size: 63

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

\[ y{\left (x \right )} = C_{2} \left (x + \frac {16 \left (x - 3\right )^{4}}{27} + \frac {28 \left (x - 3\right )^{3}}{27} + \frac {4 \left (x - 3\right )^{2}}{3} - 3\right ) + C_{1} \left (- \frac {4 \left (x - 3\right )^{4}}{27} - \frac {7 \left (x - 3\right )^{3}}{27} - \frac {\left (x - 3\right )^{2}}{3} + 1\right ) + O\left (x^{6}\right ) \]