60.1.522 problem 525

Internal problem ID [10536]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 525
Date solved : Monday, January 27, 2025 at 08:52:13 PM
CAS classification : [_separable]

\begin{align*} {y^{\prime }}^{2}-a x y y^{\prime }+2 a y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 98

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

\begin{align*} y &= 0 \\ y &= c_{1} {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{-2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{\frac {x \left (a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\ y &= c_{1} {\left (a x \,\operatorname {csgn}\left (a \right )+\sqrt {a \left (a \,x^{2}-8\right )}\right )}^{2 \,\operatorname {csgn}\left (a \right )} {\mathrm e}^{-\frac {x \left (-a x +\sqrt {a \left (a \,x^{2}-8\right )}\right )}{4}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.496 (sec). Leaf size: 195

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

\begin{align*} y(x)\to c_1 \exp \left (\frac {-8 \sqrt {a x^2-8} \arcsin \left (\frac {\sqrt {a} x}{2 \sqrt {2}}\right )+a x^2 \sqrt {8-a x^2}-\sqrt {a} x \sqrt {-\left (a x^2-8\right )^2}}{4 \sqrt {8-a x^2}}\right ) \\ y(x)\to c_1 \exp \left (\frac {8 \sqrt {a x^2-8} \arcsin \left (\frac {\sqrt {a} x}{2 \sqrt {2}}\right )+a x^2 \sqrt {8-a x^2}+\sqrt {a} x \sqrt {-\left (a x^2-8\right )^2}}{4 \sqrt {8-a x^2}}\right ) \\ y(x)\to 0 \\ \end{align*}