87.26.24 problem 33

Internal problem ID [23858]
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 253
Problem number : 33
Date solved : Thursday, October 02, 2025 at 09:45:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 \left (x +3\right )^{2} y^{\prime \prime }-\left (x^{2}+5 x +6\right ) y^{\prime }-y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 49

Order:=6; 
ode:=2*(x+3)^2*diff(diff(y(x),x),x)-(x^2+5*x+6)*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=-3);
 

\[ y = \frac {c_1 \left (1+\frac {1}{2} \left (x +3\right )+\frac {1}{8} \left (x +3\right )^{2}+\frac {1}{48} \left (x +3\right )^{3}+\frac {1}{384} \left (x +3\right )^{4}+\frac {1}{3840} \left (x +3\right )^{5}+\operatorname {O}\left (\left (x +3\right )^{6}\right )\right )}{\sqrt {x +3}}+c_2 \left (x +3\right ) \left (1+\frac {1}{5} \left (x +3\right )+\frac {1}{35} \left (x +3\right )^{2}+\frac {1}{315} \left (x +3\right )^{3}+\frac {1}{3465} \left (x +3\right )^{4}+\frac {1}{45045} \left (x +3\right )^{5}+\operatorname {O}\left (\left (x +3\right )^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 110

ode=2*(x+3)^2*D[y[x],{x,2}]-(x^2+5*x+6)*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-3,5}]
 

\[ y(x)\to c_1 (x+3) \left (\frac {(x+3)^5}{45045}+\frac {(x+3)^4}{3465}+\frac {1}{315} (x+3)^3+\frac {1}{35} (x+3)^2+\frac {x+3}{5}+1\right )+\frac {c_2 \left (\frac {(x+3)^5}{3840}+\frac {1}{384} (x+3)^4+\frac {1}{48} (x+3)^3+\frac {1}{8} (x+3)^2+\frac {x+3}{2}+1\right )}{\sqrt {x+3}} \]

✓ Sympy. Time used: 0.577 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*(x + 3)**2*Derivative(y(x), (x, 2)) - (x**2 + 5*x + 6)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=-3,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2} \left (x + 3\right ) \left (693 x + \left (x + 3\right )^{4} + 11 \left (x + 3\right )^{3} + 99 \left (x + 3\right )^{2} + 5544\right )}{887040} + \frac {C_{1} \left (1920 x + \left (x + 3\right )^{5} + 10 \left (x + 3\right )^{4} + 80 \left (x + 3\right )^{3} + 480 \left (x + 3\right )^{2} + 9600\right )}{887040 \sqrt {x + 3}} + O\left (x^{6}\right ) \]