12.12.24 problem 26

Internal problem ID [1878]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 26
Date solved : Tuesday, March 04, 2025 at 01:45:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (x +1\right ) y^{\prime }+60 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -1 \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=3\\ y^{\prime }\left (-1\right )&=-3 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
ode:=(2*x^2+4*x+5)*diff(diff(y(x),x),x)-20*(1+x)*diff(y(x),x)+60*y(x) = 0; 
ic:=y(-1) = 3, D(y)(-1) = -3; 
dsolve([ode,ic],y(x),type='series',x=-1);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 42

ode=(2*x^2+4*x+5)*D[y[x],{x,2}]-20*(x+1)*D[y[x],x]+60*y[x]==0; 
ic={y[-1]==3,Derivative[1][y][-1]==-3}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to -\frac {4}{3} (x+1)^5+20 (x+1)^4+\frac {20}{3} (x+1)^3-30 (x+1)^2-3 (x+1)+3 \]

✓ Sympy. Time used: 0.859 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-20*x - 20)*Derivative(y(x), x) + (2*x**2 + 4*x + 5)*Derivative(y(x), (x, 2)) + 60*y(x),0) 
ics = {y(-1): 3, Subs(Derivative(y(x), x), x, -1): -3} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {20 \left (x + 1\right )^{4}}{3} - 10 \left (x + 1\right )^{2} + 1\right ) + C_{1} \left (x - \frac {20 \left (x + 1\right )^{3}}{9} + 1\right ) + O\left (x^{6}\right ) \]