75.8.9 problem 207
Internal
problem
ID
[16825]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Section
8.
First
order
not
solved
for
the
derivative.
Exercises
page
67
Problem
number
:
207
Date
solved
:
Tuesday, January 28, 2025 at 09:34:01 AM
CAS
classification
:
[[_homogeneous, `class G`]]
\begin{align*} {y^{\prime }}^{2}-4 x y^{\prime }+2 y+2 x^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.069 (sec). Leaf size: 77
dsolve(diff(y(x),x)^2-4*x*diff(y(x),x)+2*y(x)+2*x^2=0,y(x), singsol=all)
\begin{align*}
y &= x^{2} \\
y &= \frac {1}{2} x^{2}+c_{1} x -\frac {1}{2} c_{1}^{2} \\
y &= \frac {1}{2} x^{2}-c_{1} x -\frac {1}{2} c_{1}^{2} \\
y &= \frac {1}{2} x^{2}-c_{1} x -\frac {1}{2} c_{1}^{2} \\
y &= \frac {1}{2} x^{2}+c_{1} x -\frac {1}{2} c_{1}^{2} \\
\end{align*}
✓ Solution by Mathematica
Time used: 14.580 (sec). Leaf size: 502
DSolve[D[y[x],x]^2-4*x*D[y[x],x]+2*y[x]+2*x^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
\text {Solve}\left [\frac {-\frac {\left (-y(x)+\sqrt {2} \sqrt {y(x)+1}-1\right ) \text {arctanh}\left (\frac {y(x)+x \left (\sqrt {2} \sqrt {y(x)+1}-1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}-\frac {\left (y(x)+\sqrt {2} \sqrt {y(x)+1}+1\right ) \text {arctanh}\left (\frac {y(x)-x \left (\sqrt {2} \sqrt {y(x)+1}+1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}+\sqrt {y(x)+1} \log \left (-x^2+2 y(x)+2 x+1\right )}{2 \sqrt {y(x)+1}}&=c_1,y(x)\right ] \\
\text {Solve}\left [\frac {\frac {\left (-y(x)+\sqrt {2} \sqrt {y(x)+1}-1\right ) \text {arctanh}\left (\frac {y(x)+x \left (\sqrt {2} \sqrt {y(x)+1}-1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)-2 \sqrt {2} \sqrt {y(x)+1}+3}}+\frac {\left (y(x)+\sqrt {2} \sqrt {y(x)+1}+1\right ) \text {arctanh}\left (\frac {y(x)-x \left (\sqrt {2} \sqrt {y(x)+1}+1\right )}{\sqrt {x^2-y(x)} \sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}\right )}{\sqrt {y(x)+2 \sqrt {2} \sqrt {y(x)+1}+3}}+\sqrt {y(x)+1} \log \left (-x^2+2 y(x)+2 x+1\right )}{2 \sqrt {y(x)+1}}&=c_1,y(x)\right ] \\
y(x)\to x^2 \\
\end{align*}