78.3.2 problem 1.b

Internal problem ID [20998]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 1.b
Date solved : Thursday, October 02, 2025 at 07:01:16 PM
CAS classification : [_linear]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 14

Order:=5; 
ode:=diff(y(x),x) = 2*x*y(x)-x^3; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x),type='series',x=0);
 

\[ y = 1+x^{2}+\frac {1}{4} x^{4}+\operatorname {O}\left (x^{5}\right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 15

ode=D[y[x],x]==2*x*y[x]-x^3; 
ic={y[0]==1}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,4}]
 

\[ y(x)\to \frac {x^4}{4}+x^2+1 \]

✓ Sympy. Time used: 0.185 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = 1 + x^{2} + \frac {x^{4}}{4} + O\left (x^{6}\right ) \]