31.6.15 problem 15
Internal
problem
ID
[5764]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
15
Date
solved
:
Tuesday, January 28, 2025 at 02:47:15 PM
CAS
classification
:
[[_homogeneous, `class A`], _dAlembert]
\begin{align*} y&=x y^{\prime }+a x \sqrt {1+{y^{\prime }}^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.112 (sec). Leaf size: 342
dsolve(y(x)=x*diff(y(x),x)+a*x*sqrt(1+(diff(y(x),x))^2),y(x), singsol=all)
\begin{align*}
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a +y \left (x \right )}{x \left (a^{2}-1\right )}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}+2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\frac {x \sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{-\frac {\operatorname {arcsinh}\left (\frac {\sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a -y \left (x \right )}{x \left (a^{2}-1\right )}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+a^{2} y \left (x \right )^{2}-2 \sqrt {-a^{2} x^{2}+x^{2}+y \left (x \right )^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.701 (sec). Leaf size: 223
DSolve[y[x]==x*D[y[x],x]+a*x*Sqrt[1+(D[y[x],x])^2],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )-2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\text {Solve}\left [\frac {-2 i \arctan \left (\frac {y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+2 i a \arctan \left (\frac {a y(x)}{x \sqrt {a^2-\frac {y(x)^2}{x^2}-1}}\right )+a \log \left (\frac {y(x)^2}{x^2}+1\right )}{2 a^2-2}&=\frac {a \log \left (x-a^2 x\right )}{1-a^2}+c_1,y(x)\right ] \\
\end{align*}