46.1.24 problem 22

Internal problem ID [9527]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. Section 6.2 SOLUTIONS ABOUT ORDINARY POINTS. EXERCISES 6.2. Page 246
Problem number : 22
Date solved : Tuesday, September 30, 2025 at 06:20:05 PM
CAS classification : [[_2nd_order, _missing_y]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 16

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

\[ y = x -\frac {1}{3} x^{3}+\frac {1}{5} x^{5}-\frac {1}{7} x^{7}+\operatorname {O}\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 26

ode=(x^2+1)*D[y[x],{x,2}]+2*x*D[y[x],x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to -\frac {x^7}{7}+\frac {x^5}{5}-\frac {x^3}{3}+x \]

✓ Sympy. Time used: 0.229 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{2} x \left (\frac {x^{4}}{5} - \frac {x^{2}}{3} + 1\right ) + C_{1} + O\left (x^{8}\right ) \]