52.4.13 problem 21

Internal problem ID [8323]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. CHAPTER 6 IN REVIEW. Page 271
Problem number : 21
Date solved : Sunday, March 30, 2025 at 12:52:01 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+x^{2} y^{\prime }+2 x y&=10 x^{3}-2 x +5 \end{align*}

Using series method with expansion around

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 54

Order:=8; 
ode:=diff(diff(y(x),x),x)+x^2*diff(y(x),x)+2*x*y(x) = 10*x^3-2*x+5; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{3} x^{3}+\frac {1}{18} x^{6}\right ) y \left (0\right )+\left (x -\frac {1}{4} x^{4}+\frac {1}{28} x^{7}\right ) y^{\prime }\left (0\right )+\frac {5 x^{2}}{2}-\frac {x^{3}}{3}+\frac {x^{6}}{18}+O\left (x^{8}\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 63

ode=D[y[x],{x,2}]+x^2*D[y[x],x]+2*x*y[x]==5-2*x+10*x^3; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,7}]
 

\[ y(x)\to \frac {x^6}{18}-\frac {x^3}{3}+\frac {5 x^2}{2}+c_2 \left (\frac {x^7}{28}-\frac {x^4}{4}+x\right )+c_1 \left (\frac {x^6}{18}-\frac {x^3}{3}+1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-10*x**3 + x**2*Derivative(y(x), x) + 2*x*y(x) + 2*x + Derivative(y(x), (x, 2)) - 5,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=8)
 

ValueError : ODE -10*x**3 + x**2*Derivative(y(x), x) + 2*x*y(x) + 2*x + Derivative(y(x), (x, 2)) - 5 does not match hint 2nd_power_series_regular