59.1.9 problem 5.4 (iv)

Internal problem ID [14993]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 5, Trivial differential equations. Exercises page 33
Problem number : 5.4 (iv)
Date solved : Thursday, October 02, 2025 at 09:58:09 AM
CAS classification : [_quadrature]

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

With initial conditions

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 13

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

\[ V = \frac {x^{2}}{2}+\ln \left (x \right )+\frac {1}{2} \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 18

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

\begin{align*} V(x)&\to \frac {1}{2} \left (x^2+2 \log (x)+1\right ) \end{align*}

✓ Sympy. Time used: 0.153 (sec). Leaf size: 14

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

\[ V{\left (x \right )} = \frac {x^{2}}{2} + \log {\left (x \right )} + \frac {1}{2} \]