54.8.5 problem 6

Internal problem ID [8667]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page 391
Problem number : 6
Date solved : Wednesday, March 05, 2025 at 06:11:59 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+x y^{\prime }+\left (x^{4}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 70

Order:=8; 
ode:=x*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^4+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_{1} x \left (1-x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{24} x^{4}-\frac {1}{24} x^{5}+\frac {31}{1008} x^{6}-\frac {47}{3528} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-x +x^{2}-\frac {1}{2} x^{3}+\frac {1}{6} x^{4}-\frac {1}{24} x^{5}+\frac {1}{24} x^{6}-\frac {31}{1008} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (1-x +\frac {1}{4} x^{3}-\frac {5}{36} x^{4}-\frac {7}{1440} x^{5}+\frac {49}{2400} x^{6}+\frac {10847}{2116800} x^{7}+\operatorname {O}\left (x^{8}\right )\right )\right ) \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 114

ode=x*D[y[x],{x,2}]+x*D[y[x],x]+(1+x^4)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to c_1 \left (\frac {1}{24} x \left (x^5-x^4+4 x^3-12 x^2+24 x-24\right ) \log (x)+\frac {-153 x^6+265 x^5-2200 x^4+5400 x^3-7200 x^2+7200}{7200}\right )+c_2 \left (\frac {31 x^7}{1008}-\frac {x^6}{24}+\frac {x^5}{24}-\frac {x^4}{6}+\frac {x^3}{2}-x^2+x\right ) \]

✓ Sympy. Time used: 0.840 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), x) + x*Derivative(y(x), (x, 2)) + (x**4 + 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

\[ y{\left (x \right )} = C_{1} x \left (\frac {31 x^{6}}{1008} - \frac {x^{5}}{24} + \frac {x^{4}}{24} - \frac {x^{3}}{6} + \frac {x^{2}}{2} - x + 1\right ) + O\left (x^{8}\right ) \]