2.1.6 problem 6

Internal problem ID [656]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.2. Integrals as general and particular solutions. Page 16
Problem number : 6
Date solved : Tuesday, September 30, 2025 at 04:05:02 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=x \sqrt {x^{2}+9} \end{align*}

With initial conditions

\begin{align*} y \left (-4\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 27

ode:=diff(y(x),x) = x*(x^2+9)^(1/2); 
ic:=[y(-4) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\sqrt {x^{2}+9}\, x^{2}}{3}+3 \sqrt {x^{2}+9}-\frac {125}{3} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 20

ode=D[y[x],x] == x*(x^2+9)^(1/2); 
ic=y[-4]==0; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{3} \left (\left (x^2+9\right )^{3/2}-125\right ) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*sqrt(x**2 + 9) + Derivative(y(x), x),0) 
ics = {y(-4): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2} \sqrt {x^{2} + 9}}{3} + 3 \sqrt {x^{2} + 9} - \frac {125}{3} \]