83.22.16 problem 16
Internal
problem
ID
[19270]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree.
Exercise
IV
(E)
at
page
63
Problem
number
:
16
Date
solved
:
Tuesday, January 28, 2025 at 01:18:14 PM
CAS
classification
:
[_dAlembert]
\begin{align*} x +y^{\prime } y&=a {y^{\prime }}^{2} \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 269
dsolve(x+y(x)*diff(y(x),x)=a*diff(y(x),x)^2,y(x), singsol=all)
\begin{align*}
\frac {-\frac {\sqrt {2}\, \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \operatorname {arcsinh}\left (\frac {y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}}{2 a}\right )}{2}+x \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}}}+c_{1} y \left (x \right )+c_{1} \sqrt {4 a x +y \left (x \right )^{2}}}{\sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+y \left (x \right )^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\
\frac {-\frac {\sqrt {2}\, \left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right ) \operatorname {arcsinh}\left (\frac {y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}}{2 a}\right )}{2}-\frac {c_{1} \sqrt {2}\, y \left (x \right )}{2}+\frac {c_{1} \sqrt {2}\, \sqrt {4 a x +y \left (x \right )^{2}}}{2}+x \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}}}}{\sqrt {\frac {-y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+y \left (x \right )^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.509 (sec). Leaf size: 57
DSolve[x+y[x]*D[y[x],x]==a*D[y[x],x]^2,y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\left \{x=\frac {a K[1] \text {arcsinh}(K[1])}{\sqrt {K[1]^2+1}}+\frac {c_1 K[1]}{\sqrt {K[1]^2+1}},y(x)=a K[1]-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ]
\]