6.13.43 problem 42

Internal problem ID [1934]
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 : 42
Date solved : Tuesday, September 30, 2025 at 05:21:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

\[ y = -3+5 x -5 x^{2}+\frac {23}{6} x^{3}-\frac {23}{12} x^{4}+\frac {11}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to \frac {11 x^5}{30}-\frac {23 x^4}{12}+\frac {23 x^3}{6}-5 x^2+5 x-3 \]

✓ Sympy. Time used: 0.289 (sec). Leaf size: 37

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

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{12} - \frac {x^{3}}{6} + 1\right ) + C_{1} x \left (- \frac {x^{3}}{3} + \frac {2 x^{2}}{3} - x + 1\right ) + O\left (x^{6}\right ) \]