77.1.59 problem 78 (page 120)

Internal problem ID [17949]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 78 (page 120)
Date solved : Tuesday, January 28, 2025 at 11:15:05 AM
CAS classification : [_dAlembert]

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

✓ Solution by Maple

Time used: 0.174 (sec). Leaf size: 391

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

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

✓ Solution by Mathematica

Time used: 0.754 (sec). Leaf size: 61

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

\[ \text {Solve}\left [\left \{x=\frac {a K[1] \arcsin (K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ] \]