15.23.9 problem 9

Internal problem ID [3359]
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 : 9
Date solved : Tuesday, March 04, 2025 at 04:36:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 3 x^{2} y^{\prime \prime }+\left (-x^{2}+5 x \right ) y^{\prime }+\left (2 x^{2}-1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 48

Order:=6; 
ode:=3*x^2*diff(diff(y(x),x),x)+(-x^2+5*x)*diff(y(x),x)+(2*x^2-1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y \left (x \right ) = \frac {c_{2} x^{{4}/{3}} \left (1+\frac {1}{21} x -\frac {61}{630} x^{2}-\frac {607}{73710} x^{3}+\frac {2297}{884520} x^{4}+\frac {14713}{50417640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{1} \left (1+x -\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{48} x^{4}+\frac {19}{2640} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 84

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

\[ y(x)\to c_1 \sqrt [3]{x} \left (\frac {14713 x^5}{50417640}+\frac {2297 x^4}{884520}-\frac {607 x^3}{73710}-\frac {61 x^2}{630}+\frac {x}{21}+1\right )+\frac {c_2 \left (\frac {19 x^5}{2640}+\frac {x^4}{48}-\frac {x^3}{6}-\frac {x^2}{2}+x+1\right )}{x} \]

✓ Sympy. Time used: 0.990 (sec). Leaf size: 73

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x**2*Derivative(y(x), (x, 2)) + (-x**2 + 5*x)*Derivative(y(x), x) + (2*x**2 - 1)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \sqrt [3]{x} \left (\frac {2297 x^{4}}{884520} - \frac {607 x^{3}}{73710} - \frac {61 x^{2}}{630} + \frac {x}{21} + 1\right ) + \frac {C_{1} \left (- \frac {17 x^{6}}{110880} + \frac {19 x^{5}}{2640} + \frac {x^{4}}{48} - \frac {x^{3}}{6} - \frac {x^{2}}{2} + x + 1\right )}{x} + O\left (x^{6}\right ) \]