65.4.17 problem 17

Internal problem ID [15676]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.2, page 53
Problem number : 17
Date solved : Thursday, October 02, 2025 at 10:22:50 AM
CAS classification : [_linear]

\begin{align*} y^{\prime }&=y x +\frac {1}{x^{2}+1} \end{align*}

With initial conditions

\begin{align*} y \left (-5\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 31

ode:=diff(y(x),x) = x*y(x)+1/(x^2+1); 
ic:=[y(-5) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \int _{-5}^{x}\frac {{\mathrm e}^{-\frac {\textit {\_z1}^{2}}{2}}}{\textit {\_z1}^{2}+1}d \textit {\_z1} {\mathrm e}^{\frac {x^{2}}{2}} \]

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 41

ode=D[y[x],x]==x*y[x]+1/(1+x^2); 
ic={y[-5]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{\frac {x^2}{2}} \int _{-5}^x\frac {e^{-\frac {1}{2} K[1]^2}}{K[1]^2+1}dK[1] \end{align*}

✓ Sympy. Time used: 22.688 (sec). Leaf size: 131

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x) + Derivative(y(x), x) - 1/(x**2 + 1),0) 
ics = {y(-5): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \int \frac {e^{- \frac {x^{2}}{2}}}{x^{2} + 1}\, dx - \int \frac {x y{\left (x \right )} e^{- \frac {x^{2}}{2}}}{x^{2} + 1}\, dx - \int \frac {x^{3} y{\left (x \right )} e^{- \frac {x^{2}}{2}}}{x^{2} + 1}\, dx = - \int \limits ^{-5} \frac {x y{\left (x \right )}}{x^{2} e^{\frac {x^{2}}{2}} + e^{\frac {x^{2}}{2}}}\, dx - \int \limits ^{-5} \frac {x^{3} y{\left (x \right )}}{x^{2} e^{\frac {x^{2}}{2}} + e^{\frac {x^{2}}{2}}}\, dx - \int \limits ^{-5} \frac {1}{x^{2} e^{\frac {x^{2}}{2}} + e^{\frac {x^{2}}{2}}}\, dx \]