69.1.4 problem 4

Internal problem ID [14157]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 06:16:20 AM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.382 (sec). Leaf size: 916

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

\begin{align*} y &= -i \left (a -x \right ) \\ y &= i \left (a -x \right ) \\ y &= -i \left (a +x \right ) \\ y &= i \left (a +x \right ) \\ y &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {-y^{2}-a^{2}+\textit {\_a}^{2}-\sqrt {\left (y^{2}+\left (a +\textit {\_a} \right )^{2}\right ) \left (y^{2}+\left (-\textit {\_a} +a \right )^{2}\right )}}{\textit {\_a} \left (\left (-y^{2}-\textit {\_a}^{2}+a^{2}\right ) \sqrt {\left (y^{2}+\left (a +\textit {\_a} \right )^{2}\right ) \left (y^{2}+\left (-\textit {\_a} +a \right )^{2}\right )}+\left (y^{2}+\left (a +\textit {\_a} \right )^{2}\right ) \left (y^{2}+\left (-\textit {\_a} +a \right )^{2}\right )\right )}d \textit {\_a} -2 \left (\int _{}^{y}\frac {4 \left (\frac {1}{4}+\left (\left (-\textit {\_f}^{2}+a^{2}-x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (\left (-\textit {\_a}^{2}-\textit {\_f}^{2}+a^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+a^{4}+\textit {\_f}^{4}\right ) \textit {\_a}}{\sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}}\, {\left (\left (-\textit {\_a}^{2}-\textit {\_f}^{2}+a^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}\right )}^{2}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (-\textit {\_f}^{2}+a^{2}-x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )}d \textit {\_f} \right )+c_{1} &= 0 \\ -\int _{\textit {\_b}}^{x}\frac {-y^{2}-a^{2}+\textit {\_a}^{2}+\sqrt {\left (y^{2}+\left (a +\textit {\_a} \right )^{2}\right ) \left (y^{2}+\left (-\textit {\_a} +a \right )^{2}\right )}}{\textit {\_a} \left (\left (y^{2}+\textit {\_a}^{2}-a^{2}\right ) \sqrt {\left (y^{2}+\left (a +\textit {\_a} \right )^{2}\right ) \left (y^{2}+\left (-\textit {\_a} +a \right )^{2}\right )}+\left (y^{2}+\left (a +\textit {\_a} \right )^{2}\right ) \left (y^{2}+\left (-\textit {\_a} +a \right )^{2}\right )\right )}d \textit {\_a} -2 \left (\int _{}^{y}\frac {4 \left (\frac {1}{4}+\left (\left (\textit {\_f}^{2}-a^{2}+x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (\left (\textit {\_a}^{2}+\textit {\_f}^{2}-a^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+a^{4}+\textit {\_f}^{4}\right ) \textit {\_a}}{\sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}}\, {\left (\left (\textit {\_a}^{2}+\textit {\_f}^{2}-a^{2}\right ) \sqrt {\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}}+\textit {\_a}^{4}+\left (2 \textit {\_f}^{2}-2 a^{2}\right ) \textit {\_a}^{2}+\left (\textit {\_f}^{2}+a^{2}\right )^{2}\right )}^{2}}d \textit {\_a} \right )\right ) \textit {\_f}}{\left (\textit {\_f}^{2}-a^{2}+x^{2}\right ) \sqrt {\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )}+\left (\textit {\_f}^{2}+\left (a -x \right )^{2}\right ) \left (\textit {\_f}^{2}+\left (a +x \right )^{2}\right )}d \textit {\_f} \right )+c_{1} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.354 (sec). Leaf size: 75

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

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