20.24.17 problem Problem 18

Internal problem ID [4002]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.2. page 739
Problem number : Problem 18
Date solved : Sunday, March 30, 2025 at 02:14:13 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+1\right ) y^{\prime \prime }+7 x y^{\prime }+2 y&=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: 0.007 (sec). Leaf size: 14

Order:=6; 
ode:=(2*x^2+1)*diff(diff(y(x),x),x)+7*x*diff(y(x),x)+2*y(x) = 0; 
ic:=y(0) = 0, D(y)(0) = 1; 
dsolve([ode,ic],y(x),type='series',x=0);
 

\[ y = x -\frac {3}{2} x^{3}+\frac {21}{8} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 19

ode=(1+2*x^2)*D[y[x],{x,2}]+7*x*D[y[x],x]+2*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {21 x^5}{8}-\frac {3 x^3}{2}+x \]

✓ Sympy. Time used: 0.852 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(7*x*Derivative(y(x), x) + (2*x**2 + 1)*Derivative(y(x), (x, 2)) + 2*y(x),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=6)
 

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