46.2.8 problem 8

Internal problem ID [9545]
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. 6.3 SOLUTIONS ABOUT SINGULAR POINTS. EXERCISES 6.3. Page 255
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 06:20:30 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.022 (sec). Leaf size: 70

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

\[ y = c_1 x \left (1-\frac {1}{2} x +\frac {1}{12} x^{2}+\frac {23}{144} x^{3}-\frac {167}{2880} x^{4}-\frac {7993}{86400} x^{5}+\frac {23599}{518400} x^{6}+\frac {1860281}{29030400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_2 \left (\ln \left (x \right ) \left (-x +\frac {1}{2} x^{2}-\frac {1}{12} x^{3}-\frac {23}{144} x^{4}+\frac {167}{2880} x^{5}+\frac {7993}{86400} x^{6}-\frac {23599}{518400} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-\frac {3}{4} x^{2}+\frac {19}{36} x^{3}+\frac {85}{1728} x^{4}-\frac {21907}{86400} x^{5}+\frac {787}{81000} x^{6}+\frac {5987917}{36288000} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 121

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

\[ y(x)\to c_1 \left (\frac {x \left (7993 x^5+5010 x^4-13800 x^3-7200 x^2+43200 x-86400\right ) \log (x)}{86400}+\frac {-107303 x^6-403755 x^5+270750 x^4+792000 x^3-1620000 x^2+1296000 x+1296000}{1296000}\right )+c_2 \left (\frac {23599 x^7}{518400}-\frac {7993 x^6}{86400}-\frac {167 x^5}{2880}+\frac {23 x^4}{144}+\frac {x^3}{12}-\frac {x^2}{2}+x\right ) \]

✓ Sympy. Time used: 0.456 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = C_{2} x + C_{1} + O\left (x^{8}\right ) \]