15.21.8 problem 30

Internal problem ID [3332]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 39, page 179
Problem number : 30
Date solved : Monday, January 27, 2025 at 07:34:22 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Solution by Maple

Time used: 0.078 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 0.124 (sec). Leaf size: 73

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

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