60.1.410 problem 412

Internal problem ID [10424]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 412
Date solved : Monday, January 27, 2025 at 07:43:06 PM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 177

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

\begin{align*} -\frac {8 \left (-\frac {3 c_{1} \left (y-\sqrt {-4 a x +y^{2}}\right ) \sqrt {\frac {-y+\sqrt {-4 a x +y^{2}}}{x}}}{8}+a x -\frac {3 y^{2}}{4}+\frac {3 y \sqrt {-4 a x +y^{2}}}{4}\right ) x}{3 \left (y-\sqrt {-4 a x +y^{2}}\right )^{2}} &= 0 \\ -\frac {8 \left (\frac {3 c_{1} \left (y+\sqrt {-4 a x +y^{2}}\right ) \sqrt {\frac {-2 y-2 \sqrt {-4 a x +y^{2}}}{x}}}{4}+a x -\frac {3 y^{2}}{4}-\frac {3 y \sqrt {-4 a x +y^{2}}}{4}\right ) x}{3 \left (y+\sqrt {-4 a x +y^{2}}\right )^{2}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.269 (sec). Leaf size: 4845

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

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