15.23.22 problem 26

Internal problem ID [3372]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 41, page 195
Problem number : 26
Date solved : Sunday, March 30, 2025 at 01:38:18 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime }-\left (x^{3}+1\right ) y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 44

Order:=6; 
ode:=2*x*diff(diff(y(x),x),x)-(x^3+1)*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{3}/{2}} \left (1-\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {52}{945} x^{3}-\frac {1049}{83160} x^{4}+\frac {5207}{5405400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \left (1+x -\frac {1}{2} x^{2}+\frac {1}{18} x^{3}+\frac {17}{360} x^{4}-\frac {377}{12600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 81

ode=2*x*D[y[x],{x,2}]-(1+x^3)*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (-\frac {377 x^5}{12600}+\frac {17 x^4}{360}+\frac {x^3}{18}-\frac {x^2}{2}+x+1\right )+c_1 \left (\frac {5207 x^5}{5405400}-\frac {1049 x^4}{83160}+\frac {52 x^3}{945}+\frac {x^2}{70}-\frac {x}{5}+1\right ) x^{3/2} \]

✓ Sympy. Time used: 0.994 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x*Derivative(y(x), (x, 2)) - (x**3 + 1)*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (- \frac {377 x^{5}}{12600} + \frac {17 x^{4}}{360} + \frac {x^{3}}{18} - \frac {x^{2}}{2} + x + 1\right ) + C_{1} x^{\frac {3}{2}} \left (\frac {52 x^{3}}{945} + \frac {x^{2}}{70} - \frac {x}{5} + 1\right ) + O\left (x^{6}\right ) \]