75.1.5 problem 6
Internal
problem
ID
[16668]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
1.
Basic
concepts
and
definitions.
Exercises
page
18
Problem
number
:
6
Date
solved
:
Tuesday, January 28, 2025 at 09:16:47 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
\begin{align*} y^{\prime }&=\sqrt {x^{2}-y}-x \end{align*}
✓ Solution by Maple
Time used: 0.010 (sec). Leaf size: 171
dsolve(diff(y(x),x)=sqrt(x^2-y(x))-x,y(x), singsol=all)
\[
\frac {250 \left (x^{2}-y\right )^{{3}/{2}} \left (x^{2}+4 y\right ) \left (x^{6} c_{1} y^{2}+\frac {12 x^{4} c_{1} y^{3}}{5}+\frac {48 x^{2} c_{1} y^{4}}{25}+\frac {64 c_{1} y^{5}}{125}-\frac {1}{125}\right )-250 x \left (x^{4}+\frac {5 x^{2} y}{2}+10 y^{2}\right ) \left (x^{6} c_{1} y^{2}+\frac {12 x^{4} c_{1} y^{3}}{5}+\frac {48 x^{2} c_{1} y^{4}}{25}+\frac {64 c_{1} y^{5}}{125}+\frac {1}{125}\right )}{\left (5 x^{2}+4 y\right )^{3} y^{2} \left (2 \sqrt {x^{2}-y}+3 x \right )^{3} \left (-\sqrt {x^{2}-y}+x \right )^{2}} = 0
\]
✓ Solution by Mathematica
Time used: 4.635 (sec). Leaf size: 416
DSolve[D[y[x],x]==Sqrt[x^2-y[x]]-x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,1\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,2\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,3\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,4\right ] \\
y(x)\to \text {Root}\left [64 \text {$\#$1}^5+240 \text {$\#$1}^4 x^2+300 \text {$\#$1}^3 x^4+\text {$\#$1}^2 \left (125 x^6-40 e^{5 c_1} x\right )-10 \text {$\#$1} e^{5 c_1} x^3-4 e^{5 c_1} x^5+e^{10 c_1}\&,5\right ] \\
y(x)\to 0 \\
\end{align*}