60.1.499 problem 502

Internal problem ID [10513]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 502
Date solved : Monday, January 27, 2025 at 08:33:29 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.373 (sec). Leaf size: 200

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

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

✓ Solution by Mathematica

Time used: 2.535 (sec). Leaf size: 71

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

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