60.2.66 problem 642
Internal
problem
ID
[10653]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
642
Date
solved
:
Tuesday, January 28, 2025 at 05:02:34 PM
CAS
classification
:
[_rational]
\begin{align*} y^{\prime }&=\frac {\left (-y^{2}+4 a x \right )^{2}}{y} \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 227
dsolve(diff(y(x),x) = (-y(x)^2+4*a*x)^2/y(x),y(x), singsol=all)
\begin{align*}
y &= \frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \sqrt {a}\, \left (c_{1} \left (\sqrt {a}\, x -\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+\sqrt {a}\, x +\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\
y &= -\frac {2 \,{\mathrm e}^{4 a \,x^{2}+2 \sqrt {2}\, \sqrt {a}\, x} \sqrt {\left (c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1\right ) \sqrt {a}\, \left (c_{1} \left (\sqrt {a}\, x -\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+\sqrt {a}\, x +\frac {\sqrt {2}}{4}\right ) {\mathrm e}^{-8 a \,x^{2}} {\mathrm e}^{-4 \sqrt {2}\, \sqrt {a}\, x}}}{c_{1} {\mathrm e}^{4 \sqrt {2}\, \sqrt {a}\, x}+1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.161 (sec). Leaf size: 147
DSolve[D[y[x],x] == (4*a*x - y[x]^2)^2/y[x],y[x],x,IncludeSingularSolutions -> True]
\[
\text {Solve}\left [\int _1^{y(x)}\left (-\frac {2 a K[2]}{K[2]^4-8 a x K[2]^2+16 a^2 x^2-2 a}-\int _1^x-\frac {4 a^2 \left (4 K[2]^3-16 a K[1] K[2]\right )}{\left (K[2]^4-8 a K[1] K[2]^2+16 a^2 K[1]^2-2 a\right )^2}dK[1]\right )dK[2]+\int _1^x\left (\frac {4 a^2}{y(x)^4-8 a K[1] y(x)^2+16 a^2 K[1]^2-2 a}+2 a\right )dK[1]=c_1,y(x)\right ]
\]