12.13.46 problem 45

Internal problem ID [1937]
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 : 45
Date solved : Saturday, March 29, 2025 at 11:43:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 20

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

\[ y = 1-2 x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {15}{8} x^{4}-\frac {71}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to -\frac {71 x^5}{60}+\frac {15 x^4}{8}-\frac {11 x^3}{6}+\frac {3 x^2}{2}-2 x+1 \]

✓ Sympy. Time used: 0.990 (sec). Leaf size: 46

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

\[ y{\left (x \right )} = C_{2} \left (- \frac {11 x^{4}}{24} + \frac {x^{3}}{2} - \frac {x^{2}}{2} + 1\right ) + C_{1} x \left (- \frac {7 x^{3}}{6} + \frac {7 x^{2}}{6} - x + 1\right ) + O\left (x^{6}\right ) \]