66.1.14 problem Problem 14

Internal problem ID [13861]
Book : Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS, MOSCOW, Third printing 1977.
Section : Chapter 1, First-Order Differential Equations. Problems page 88
Problem number : Problem 14
Date solved : Tuesday, January 28, 2025 at 06:05:55 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.078 (sec). Leaf size: 43

dsolve(x^2+diff(y(x),x)^2=1,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 57

DSolve[x^2+D[y[x],x]^2==1,y[x],x,IncludeSingularSolutions -> True]
 

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