15.25.8 problem 7

Internal problem ID [3395]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 43, page 209
Problem number : 7
Date solved : Tuesday, March 04, 2025 at 04:37:34 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 66

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

\[ y \left (x \right ) = \frac {c_{2} x^{{4}/{3}} \left (1+\frac {1}{7} x -\frac {1}{10} x^{2}+\frac {29}{2730} x^{3}-\frac {17}{87360} x^{4}-\frac {1193}{8299200} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{4}+\frac {1}{60} x -\frac {47}{960} x^{2}+\frac {673}{52800} x^{3}-\frac {1169}{316800} x^{4}+\operatorname {O}\left (x^{5}\right )\right )+c_{1} \left (1+7 x -\frac {1}{2} x^{2}-\frac {29}{30} x^{3}+\frac {73}{480} x^{4}-\frac {167}{26400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.06 (sec). Leaf size: 255

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

\[ y(x)\to \frac {c_1 \left (-\frac {167 x^5}{26400}+\frac {73 x^4}{480}-\frac {29 x^3}{30}-\frac {x^2}{2}+7 x+1\right )}{x}+c_2 \sqrt [3]{x} \left (-\frac {1193 x^5}{8299200}-\frac {17 x^4}{87360}+\frac {29 x^3}{2730}-\frac {x^2}{10}+\frac {x}{7}+1\right )+\sqrt [3]{x} \left (-\frac {1193 x^5}{8299200}-\frac {17 x^4}{87360}+\frac {29 x^3}{2730}-\frac {x^2}{10}+\frac {x}{7}+1\right ) \left (\frac {19491 x^{17/3}}{8800}-\frac {541 x^{14/3}}{256}+\frac {107 x^{11/3}}{55}-\frac {99 x^{8/3}}{64}+\frac {3 x^{5/3}}{5}+\frac {3 x^{2/3}}{8}\right )+\frac {\left (-\frac {167 x^5}{26400}+\frac {73 x^4}{480}-\frac {29 x^3}{30}-\frac {x^2}{2}+7 x+1\right ) \left (-\frac {652399 x^6}{2096640}+\frac {2039 x^5}{6825}-\frac {313 x^4}{1120}+\frac {5 x^3}{21}-\frac {x^2}{8}\right )}{x} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**3 + 3*x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(5 - x)*Derivative(y(x), 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)
 

ValueError : ODE x**3 + 3*x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(5 - x)*Derivative(y(x), x) - x + (2*x**2 - 1)*y(x) does not match hint 2nd_power_series_regular