68.1.36 problem 43

Internal problem ID [17103]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 1. Introduction to Differential Equations. Exercises 1.1, page 10
Problem number : 43
Date solved : Thursday, October 02, 2025 at 01:43:14 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\left (-x^{2}+4\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 32

ode:=diff(y(x),x) = (-x^2+4)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-x^{3}+10 x \right ) \sqrt {-x^{2}+4}}{4}+c_1 +6 \arcsin \left (\frac {x}{2}\right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 37

ode=D[y[x],x]==(4-x^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 6 \arcsin \left (\frac {x}{2}\right )-\frac {1}{4} x \sqrt {4-x^2} \left (x^2-10\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.842 (sec). Leaf size: 128

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(4 - x**2)**(3/2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + 2 x \sqrt {4 - x^{2}} - \begin {cases} \frac {i x^{5}}{4 \sqrt {x^{2} - 4}} - \frac {3 i x^{3}}{2 \sqrt {x^{2} - 4}} + \frac {2 i x}{\sqrt {x^{2} - 4}} - 2 i \operatorname {acosh}{\left (\frac {x}{2} \right )} & \text {for}\: \left |{x^{2}}\right | > 4 \\- \frac {x^{5}}{4 \sqrt {4 - x^{2}}} + \frac {3 x^{3}}{2 \sqrt {4 - x^{2}}} - \frac {2 x}{\sqrt {4 - x^{2}}} + 2 \operatorname {asin}{\left (\frac {x}{2} \right )} & \text {otherwise} \end {cases} + 8 \operatorname {asin}{\left (\frac {x}{2} \right )} \]