62.10.5 problem Ex 6

Internal problem ID [12842]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 17. Other forms which Integrating factors can be found. Page 25
Problem number : Ex 6
Date solved : Tuesday, January 28, 2025 at 04:26:34 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 59

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

\begin{align*} y &= \frac {m -\sqrt {-4 c_{1}^{2} x^{2}-4 c_{1} x +m^{2}}}{2 c_{1}} \\ y &= \frac {m +\sqrt {-4 c_{1}^{2} x^{2}-4 c_{1} x +m^{2}}}{2 c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.214 (sec). Leaf size: 54

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

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {m K[1]^2-2 K[1]-m}{(m K[1]-1) \left (K[1]^2+1\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]