Internal problem ID [5093]
Book: THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T.
CHAU, CRC Press. Boca Raton, FL. 2018
Section: Chapter 3. Ordinary Differential Equations. Section 3.2 FIRST ORDER ODE. Page
114
Problem number: Example 3.14.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [_dAlembert]
Solve \begin {gather*} \boxed {x +y y^{\prime }-a \left (y^{\prime }\right )^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.141 (sec). Leaf size: 264
dsolve(x+y(x)*diff(y(x),x)=a*(diff(y(x),x))^2,y(x), singsol=all)
\begin{align*} \frac {c_{1} \left (y \relax (x )+\sqrt {4 a x +y \relax (x )^{2}}\right )}{\sqrt {\frac {y \relax (x ) \sqrt {4 a x +y \relax (x )^{2}}+2 a^{2}+2 a x +y \relax (x )^{2}}{a^{2}}}}+x -\frac {\sqrt {2}\, \left (y \relax (x )+\sqrt {4 a x +y \relax (x )^{2}}\right ) \arcsinh \left (\frac {y \relax (x )+\sqrt {4 a x +y \relax (x )^{2}}}{2 a}\right )}{2 \sqrt {\frac {y \relax (x ) \sqrt {4 a x +y \relax (x )^{2}}+2 a^{2}+2 a x +y \relax (x )^{2}}{a^{2}}}} = 0 \\ \frac {c_{1} \left (-y \relax (x )+\sqrt {4 a x +y \relax (x )^{2}}\right )}{\sqrt {-\frac {2 \left (y \relax (x ) \sqrt {4 a x +y \relax (x )^{2}}-2 a^{2}-2 a x -y \relax (x )^{2}\right )}{a^{2}}}}+x -\frac {\left (-y \relax (x )+\sqrt {4 a x +y \relax (x )^{2}}\right ) \arcsinh \left (\frac {-y \relax (x )+\sqrt {4 a x +y \relax (x )^{2}}}{2 a}\right )}{\sqrt {-\frac {2 \left (y \relax (x ) \sqrt {4 a x +y \relax (x )^{2}}-2 a^{2}-2 a x -y \relax (x )^{2}\right )}{a^{2}}}} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.396 (sec). Leaf size: 68
DSolve[x+y[x]*y'[x]==a*(y'[x])^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\left \{x=\frac {a K[1] \tanh ^{-1}\left (\frac {K[1]}{\sqrt {K[1]^2+1}}\right )}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]