84.30.2 problem 18.12

Internal problem ID [22291]
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 : 18.12
Date solved : Thursday, October 02, 2025 at 08:37:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 2 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 67

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

\[ y = \left (1-\frac {\left (x -2\right )^{3}}{3}+\frac {\left (x -2\right )^{4}}{6}+\frac {\left (x -2\right )^{5}}{20}\right ) y \left (2\right )+\left (x -2-\frac {3 \left (x -2\right )^{2}}{2}+\left (x -2\right )^{3}-\frac {\left (x -2\right )^{4}}{6}-\frac {\left (x -2\right )^{5}}{4}\right ) y^{\prime }\left (2\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 74

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

\[ y(x)\to c_1 \left (\frac {1}{20} (x-2)^5+\frac {1}{6} (x-2)^4-\frac {1}{3} (x-2)^3+1\right )+c_2 \left (-\frac {1}{4} (x-2)^5-\frac {1}{6} (x-2)^4+(x-2)^3-\frac {3}{2} (x-2)^2+x-2\right ) \]

✓ Sympy. Time used: 0.442 (sec). Leaf size: 87

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x - 2)**2*y(x) + (x - 2)*Derivative(y(x), (x, 2)) + (3*x**2 - 9*x + 6)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=2,n=6)
 

\[ y{\left (x \right )} = - \frac {3 \left (x - 2\right )^{4} r{\left (3 \right )}}{4} - \frac {\left (x - 2\right )^{4} r{\left (2 \right )}}{2} + \frac {\left (x - 2\right )^{5} r{\left (2 \right )}}{5} + \frac {7 \left (x - 2\right )^{6} r{\left (3 \right )}}{30} + \frac {\left (x - 2\right )^{6} r{\left (2 \right )}}{15} + C_{2} \left (x + \frac {\left (x - 2\right )^{6}}{24} + \frac {\left (x - 2\right )^{5}}{20} - \frac {\left (x - 2\right )^{4}}{6} - 2\right ) + C_{1} + O\left (x^{6}\right ) \]