15.25.14 problem 13

Internal problem ID [3401]
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 : 13
Date solved : Sunday, March 30, 2025 at 01:39:14 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 63

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

\[ y = \frac {c_1 \left (1-x +\frac {1}{6} x^{2}-\frac {1}{90} x^{3}+\frac {1}{2520} x^{4}-\frac {1}{113400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_2 \left (1-\frac {1}{3} x +\frac {1}{30} x^{2}-\frac {1}{630} x^{3}+\frac {1}{22680} x^{4}-\frac {1}{1247400} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}}+x \left (\frac {1}{6}+\frac {1}{18} x +\frac {17}{504} x^{2}-\frac {17}{22680} x^{3}+\frac {17}{1496880} x^{4}+\operatorname {O}\left (x^{5}\right )\right ) \]

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 237

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

\[ y(x)\to \frac {c_1 \left (-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right )}{x}+\frac {c_2 \left (-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right )}{\sqrt {x}}+\frac {\left (-\frac {x^5}{1247400}+\frac {x^4}{22680}-\frac {x^3}{630}+\frac {x^2}{30}-\frac {x}{3}+1\right ) \left (\frac {131 x^{11/2}}{4620}-\frac {76 x^{9/2}}{405}+\frac {x^{7/2}}{21}+\frac {2 x^{3/2}}{3}\right )}{\sqrt {x}}+\frac {\left (-\frac {x^5}{113400}+\frac {x^4}{2520}-\frac {x^3}{90}+\frac {x^2}{6}-x+1\right ) \left (-\frac {103 x^6}{19440}+\frac {19 x^5}{315}-\frac {7 x^4}{40}-\frac {2 x^3}{9}-\frac {x^2}{2}\right )}{x} \]

✗ Sympy

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

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