6.13.9 problem 9

Internal problem ID [1900]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 05:21:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 20

Order:=6; 
ode:=(2*x^2+3*x)*diff(diff(y(x),x),x)+10*(1+x)*diff(y(x),x)+8*y(x) = 0; 
ic:=[y(-1) = 1, D(y)(-1) = -1]; 
dsolve([ode,op(ic)],y(x),type='series',x=-1);
 

\[ y = 1-\left (1+x \right )+4 \left (1+x \right )^{2}-\frac {13}{3} \left (1+x \right )^{3}+\frac {77}{6} \left (1+x \right )^{4}-\frac {278}{15} \left (1+x \right )^{5}+\operatorname {O}\left (\left (1+x \right )^{6}\right ) \]

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

ode=(3*x+2*x^2)*D[y[x],{x,2}]+10*(1+x)*D[y[x],x]+8*y[x]==0; 
ic={y[-1]==1,Derivative[1][y][-1]==-1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,-1,5}]
 

\[ y(x)\to -\frac {278}{15} (x+1)^5+\frac {77}{6} (x+1)^4-\frac {13}{3} (x+1)^3+4 (x+1)^2-x \]

✓ Sympy. Time used: 0.315 (sec). Leaf size: 53

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((10*x + 10)*Derivative(y(x), x) + (2*x**2 + 3*x)*Derivative(y(x), (x, 2)) + 8*y(x),0) 
ics = {y(-1): 1, Subs(Derivative(y(x), x), x, -1): -1} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=-1,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {34 \left (x + 1\right )^{4}}{3} - \frac {4 \left (x + 1\right )^{3}}{3} + 4 \left (x + 1\right )^{2} + 1\right ) + C_{1} \left (x - \frac {3 \left (x + 1\right )^{4}}{2} + 3 \left (x + 1\right )^{3} + 1\right ) + O\left (x^{6}\right ) \]