31.6.16 problem 16
Internal
problem
ID
[5765]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
16
Date
solved
:
Tuesday, January 28, 2025 at 02:52:30 PM
CAS
classification
:
[_dAlembert]
\begin{align*} x -y^{\prime } y&=a {y^{\prime }}^{2} \end{align*}
✓ Solution by Maple
Time used: 0.068 (sec). Leaf size: 396
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 \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right )}{\sqrt {\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}-2 a}{a}}\, \sqrt {\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}+2 a}{a}}}+x +\frac {\left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right ) \left (3 \ln \left (2\right )-2 \ln \left (\frac {2 \sqrt {\frac {y \left (x \right )^{2}-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x}{a^{2}}}\, a -\sqrt {2}\, \left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right )}{a}\right )\right ) \sqrt {2}}{4 \sqrt {\frac {y \left (x \right )^{2}-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x}{a^{2}}}} &= 0 \\
\frac {c_{1} \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{2 \sqrt {\frac {-y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}-2 a}{a}}\, \sqrt {\frac {-y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}+2 a}{a}}}+x -\frac {\sqrt {2}\, \left (-\frac {3 \ln \left (2\right )}{2}+\ln \left (\frac {2 \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}\, a -\sqrt {2}\, \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{a}\right )\right ) \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right )}{2 \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}-2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.785 (sec). Leaf size: 61
DSolve[x-y[x]*D[y[x],x]==a*(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{x=\frac {a K[1] \arcsin (K[1])}{\sqrt {1-K[1]^2}}+\frac {c_1 K[1]}{\sqrt {1-K[1]^2}},y(x)=\frac {x}{K[1]}-a K[1]\right \},\{y(x),K[1]\}\right ]
\]