60.1.533 problem 536

Internal problem ID [10547]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 536
Date solved : Monday, January 27, 2025 at 08:53:37 PM
CAS classification : [_quadrature]

\begin{align*} \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{3}+b x \left (-a^{2}+x^{2}\right ) {y^{\prime }}^{2}+y^{\prime }+b x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 52

dsolve((-a^2+x^2)*diff(y(x),x)^3+b*x*(-a^2+x^2)*diff(y(x),x)^2+diff(y(x),x)+b*x=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 64

DSolve[b*x + D[y[x],x] + b*x*(-a^2 + x^2)*D[y[x],x]^2 + (-a^2 + x^2)*D[y[x],x]^3==0,y[x],x,IncludeSingularSolutions -> True]
 

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