46.1.19 problem 17

Internal problem ID [9522]
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 : 17
Date solved : Tuesday, September 30, 2025 at 06:20:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 49

Order:=8; 
ode:=(x^2+2)*diff(diff(y(x),x),x)+3*x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1+\frac {1}{4} x^{2}-\frac {7}{96} x^{4}+\frac {161}{5760} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}-\frac {17}{720} x^{7}\right ) y^{\prime }\left (0\right )+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 56

ode=(x^2+2)*D[y[x],{x,2}]+3*x*D[y[x],x]-y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_2 \left (-\frac {17 x^7}{720}+\frac {7 x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {161 x^6}{5760}-\frac {7 x^4}{96}+\frac {x^2}{4}+1\right ) \]

✓ Sympy. Time used: 0.285 (sec). Leaf size: 44

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

\[ y{\left (x \right )} = C_{2} \left (\frac {161 x^{6}}{5760} - \frac {7 x^{4}}{96} + \frac {x^{2}}{4} + 1\right ) + C_{1} x \left (\frac {7 x^{4}}{120} - \frac {x^{2}}{6} + 1\right ) + O\left (x^{8}\right ) \]