10.2.20 problem 20

Internal problem ID [1148]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.2. Page 48
Problem number : 20
Date solved : Saturday, March 29, 2025 at 10:42:18 PM
CAS classification : [_separable]

\begin{align*} \sqrt {-x^{2}+1}\, y^{2} y^{\prime }&=\arcsin \left (x \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.167 (sec). Leaf size: 16

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

\[ y = \frac {\left (8+12 \arcsin \left (x \right )^{2}\right )^{{1}/{3}}}{2} \]

✓ Mathematica. Time used: 0.492 (sec). Leaf size: 19

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

\[ y(x)\to \sqrt [3]{\frac {3 \arcsin (x)^2}{2}+1} \]

✓ Sympy. Time used: 1.526 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = \sqrt [3]{\frac {3 \operatorname {asin}^{2}{\left (x \right )}}{2} + 1} \]