86.3.2 problem 2

Internal problem ID [23093]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 4. Linear equations of the first order. Exercise 4a at page 56
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:20:03 PM
CAS classification : [_separable]

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

With initial conditions

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 14

ode:=diff(y(x),x)+2*x*y(x) = x; 
ic:=[y(0) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {1}{2}+\frac {{\mathrm e}^{-x^{2}}}{2} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 18

ode=D[y[x],x]+2*x*y[x]==x; 
ic={y[0]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} \left (e^{-x^2}+1\right ) \end{align*}

✓ Sympy. Time used: 0.200 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = \frac {1}{2} + \frac {e^{- x^{2}}}{2} \]