29.34.4 problem 999

Internal problem ID [5575]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 34
Problem number : 999
Date solved : Monday, January 27, 2025 at 12:10:11 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} a^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right ) {y^{\prime }}^{2}+2 a \,b^{2} c y^{\prime }+c^{2} \left (b^{2}-\left (c x -a y\right )^{2}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.477 (sec). Leaf size: 200

dsolve(a^2*(b^2-(c*x-a*y(x))^2)*diff(y(x),x)^2+2*a*b^2*c*diff(y(x),x)+c^2*(b^2-(c*x-a*y(x))^2) = 0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= \frac {c x -\sqrt {2}\, b}{a} \\ y \left (x \right ) &= \frac {c x +\sqrt {2}\, b}{a} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-a \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2} a^{2}-2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}}{\textit {\_a}^{2} a^{2}-2 b^{2}}d \textit {\_a} \right )+2 c c_{1} -2 c x \right ) a +c x}{a} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2} a^{2}-2 b^{2}-\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}}{\textit {\_a}^{2} a^{2}-2 b^{2}}d \textit {\_a} \right )+2 c c_{1} -2 c x \right ) a +c x}{a} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.481 (sec). Leaf size: 71

DSolve[a^2 ( b^2 -(c x-a y[x])^2 ) (D[y[x],x])^2 +2 a b^2 c D[y[x],x]+c^2(b^2-(c x-a y[x])^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {c c_1-\sqrt {b^2-c^2 (x-c_1){}^2}}{a} \\ y(x)\to \frac {\sqrt {b^2-c^2 (x-c_1){}^2}+c c_1}{a} \\ \end{align*}