60.1.389 problem 390
Internal
problem
ID
[10403]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
390
Date
solved
:
Monday, January 27, 2025 at 07:40:00 PM
CAS
classification
:
[_dAlembert]
\begin{align*} {y^{\prime }}^{2}+a y y^{\prime }-b x -c&=0 \end{align*}
✓ Solution by Maple
Time used: 0.049 (sec). Leaf size: 291
dsolve(diff(y(x),x)^2+a*y(x)*diff(y(x),x)-b*x-c = 0,y(x), singsol=all)
\[
y = \frac {4 \left (a x b +a c -\frac {1}{2} b \right ) {\mathrm e}^{\operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-a \,{\mathrm e}^{2 \textit {\_Z}} b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -a \,{\mathrm e}^{2 \textit {\_Z}} c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}-b^{2} {\mathrm e}^{-\operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-a \,{\mathrm e}^{2 \textit {\_Z}} b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -a \,{\mathrm e}^{2 \textit {\_Z}} c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}-{\mathrm e}^{3 \operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-a \,{\mathrm e}^{2 \textit {\_Z}} b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -a \,{\mathrm e}^{2 \textit {\_Z}} c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}}{a^{{3}/{2}} \left (2 \,{\mathrm e}^{2 \operatorname {RootOf}\left (\sqrt {a}\, c_{1} b \,{\mathrm e}^{2 \textit {\_Z}}-a \,{\mathrm e}^{2 \textit {\_Z}} b x -{\mathrm e}^{2 \textit {\_Z}} \textit {\_Z} b -a \,{\mathrm e}^{2 \textit {\_Z}} c +\sqrt {a}\, c_{1} b^{2}+a \,b^{2} x -\textit {\_Z} \,b^{2}+a b c \right )}+2 b \right )}
\]
✓ Solution by Mathematica
Time used: 0.956 (sec). Leaf size: 193
DSolve[-c - b*x + a*y[x]*D[y[x],x] + D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{x=\exp \left (\int _1^{K[1]}-\frac {b}{a K[2]^2 \left (K[2]-\frac {b}{a K[2]}\right )}dK[2]\right ) \int \frac {\left (-\frac {c-K[1]^2}{a K[1]^2}-\frac {2}{a}\right ) \exp \left (-\int _1^{K[1]}-\frac {b}{a K[2]^2 \left (K[2]-\frac {b}{a K[2]}\right )}dK[2]\right )}{K[1]-\frac {b}{a K[1]}} \, dK[1]+c_1 \exp \left (\int _1^{K[1]}-\frac {b}{a K[2]^2 \left (K[2]-\frac {b}{a K[2]}\right )}dK[2]\right ),y(x)=\frac {b x}{a K[1]}+\frac {c-K[1]^2}{a K[1]}\right \},\{y(x),K[1]\}\right ]
\]