Internal problem ID [7970]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 390.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
Solve \begin {gather*} \boxed {\left (y^{\prime }\right )^{2}+a y y^{\prime }-b x -c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.245 (sec). Leaf size: 416
dsolve(diff(y(x),x)^2+a*y(x)*diff(y(x),x)-b*x-c = 0,y(x), singsol=all)
\[ y \relax (x ) = \frac {2 b x \,{\mathrm e}^{\RootOf \left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}}{\left ({\mathrm e}^{2 \RootOf \left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}+b \right ) \sqrt {a}}+\frac {2 \left (-\frac {\left ({\mathrm e}^{2 \RootOf \left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}+b \right )^{2} {\mathrm e}^{-2 \RootOf \left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}}{4 a}+c \right ) {\mathrm e}^{\RootOf \left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}}{\left ({\mathrm e}^{2 \RootOf \left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{2 \textit {\_Z}} a b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -{\mathrm e}^{2 \textit {\_Z}} a c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}+b \right ) \sqrt {a}} \]
✓ Solution by Mathematica
Time used: 2.194 (sec). Leaf size: 161
DSolve[-c - b*x + a*y[x]*y'[x] + y'[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\left (\frac {a \log \left (\sqrt {b-a K[1]^2}-\sqrt {-a} K[1]\right )}{(-a)^{3/2}}-\frac {c \sqrt {b-a K[1]^2}}{b K[1]}\right ) \exp \left (b \left (\frac {\log (K[1])}{b}-\frac {\log \left (b-a K[1]^2\right )}{2 b}\right )\right )+c_1 \exp \left (b \left (\frac {\log (K[1])}{b}-\frac {\log \left (b-a K[1]^2\right )}{2 b}\right )\right ),y(x)=\frac {b x}{a K[1]}+\frac {c-K[1]^2}{a K[1]}\right \},\{y(x),K[1]\}\right ] \]