Internal problem ID [8760]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 424.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _dAlembert]
\[ \boxed {x {y^{\prime }}^{2}+a y y^{\prime }=-b x} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 224
dsolve(x*diff(y(x),x)^2+a*y(x)*diff(y(x),x)+b*x = 0,y(x), singsol=all)
\begin{align*} \frac {c_{1} \left (a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\right ) {\left (\frac {a \left (a^{2} y \left (x \right )^{2}+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\, a y \left (x \right )+a y \left (x \right )^{2}-2 b \,x^{2}+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\, y \left (x \right )\right )}{2 x^{2}}\right )}^{-\frac {a +2}{2 \left (a +1\right )}}}{x}+x = 0 \frac {\left (-a y \left (x \right )+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\right ) c_{1} {\left (-\frac {a \left (-a^{2} y \left (x \right )^{2}+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\, a y \left (x \right )-a y \left (x \right )^{2}+2 b \,x^{2}+\sqrt {a^{2} y \left (x \right )^{2}-4 b \,x^{2}}\, y \left (x \right )\right )}{2 x^{2}}\right )}^{-\frac {a +2}{2 \left (a +1\right )}}}{x}+x = 0 \end{align*}
✓ Solution by Mathematica
Time used: 2.076 (sec). Leaf size: 423
DSolve[b*x + a*y[x]*y'[x] + x*y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [-\frac {i \left (2 \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 (a+1) \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}-\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}=c_1-\frac {1}{2} i \log (x),y(x)\right ] \text {Solve}\left [\frac {i \left (2 (a+1) \log \left (-i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}+2 i \sqrt {b}\right )+2 \log \left (i \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {a y(x)}{x}-2 i \sqrt {b}\right )-(a+2) \log \left (-\frac {i (a+2) y(x) \sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}}{x}+2 \sqrt {b} \left (\sqrt {4 b-\frac {a^2 y(x)^2}{x^2}}+\frac {i (a+2) y(x)}{x}\right )+\frac {a^2 y(x)^2}{x^2}-4 b\right )\right )}{4 (a+1)}=\frac {1}{2} i \log (x)+c_1,y(x)\right ] \end{align*}