60.1.546 problem 549
Internal
problem
ID
[10560]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
549
Date
solved
:
Monday, January 27, 2025 at 09:05:14 PM
CAS
classification
:
[_quadrature]
\begin{align*} x^{2} \left (1+{y^{\prime }}^{2}\right )^{3}-a^{2}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 585
dsolve(x^2*(diff(y(x),x)^2+1)^3-a^2=0,y(x), singsol=all)
\begin{align*}
y &= \frac {-\sqrt {\frac {x \left (a^{2} x \right )^{{1}/{3}} \left (a^{2}-\left (a^{2} x \right )^{{2}/{3}}\right )}{a^{2}}}\, a^{2}+c_{1} \left (a^{2} x \right )^{{2}/{3}}+\left (a^{2} x \right )^{{2}/{3}} \sqrt {\frac {x \left (a^{2} x \right )^{{1}/{3}} \left (a^{2}-\left (a^{2} x \right )^{{2}/{3}}\right )}{a^{2}}}}{\left (a^{2} x \right )^{{2}/{3}}} \\
y &= \frac {\left (a^{2}-\left (a^{2} x \right )^{{2}/{3}}\right ) \sqrt {\frac {x \left (a^{2} x \right )^{{1}/{3}} \left (a^{2}-\left (a^{2} x \right )^{{2}/{3}}\right )}{a^{2}}}+c_{1} \left (a^{2} x \right )^{{2}/{3}}}{\left (a^{2} x \right )^{{2}/{3}}} \\
y &= -\frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{{1}/{3}}+\left (a^{2} x \right )^{{1}/{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{{1}/{3}} \left (2 i \left (a^{2} x \right )^{{2}/{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{{2}/{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{{1}/{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{{1}/{3}}\right ) x}\, \left (a^{2} x \right )^{{2}/{3}}}+c_{1} \\
y &= \frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (a^{2} x \right )^{{1}/{3}}+\left (a^{2} x \right )^{{1}/{3}}+2 x \right )}\, \sqrt {\frac {x \left (a^{2} x \right )^{{1}/{3}} \left (2 i \left (a^{2} x \right )^{{2}/{3}}+i a^{2}-\sqrt {3}\, a^{2}\right )}{a^{2}}}\, \left (2 \left (a^{2} x \right )^{{2}/{3}}+a^{2}+i \sqrt {3}\, a^{2}\right )}{4 \sqrt {\left (i \left (a^{2} x \right )^{{1}/{3}}+2 i x -\sqrt {3}\, \left (a^{2} x \right )^{{1}/{3}}\right ) x}\, \left (a^{2} x \right )^{{2}/{3}}}+c_{1} \\
y &= \frac {\sqrt {2}\, \left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{{2}/{3}}\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{{2}/{3}}\right ) x \left (a^{2} x \right )^{{1}/{3}}}{a^{2}}}+4 c_{1} \left (a^{2} x \right )^{{2}/{3}}}{4 \left (a^{2} x \right )^{{2}/{3}}} \\
y &= -\frac {\left (-2 \left (a^{2} x \right )^{{2}/{3}} \sqrt {2}+a^{2} \left (i \sqrt {6}-\sqrt {2}\right )\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (a^{2} x \right )^{{2}/{3}}\right ) x \left (a^{2} x \right )^{{1}/{3}}}{a^{2}}}-4 c_{1} \left (a^{2} x \right )^{{2}/{3}}}{4 \left (a^{2} x \right )^{{2}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.116 (sec). Leaf size: 216
DSolve[-a^2 + x^2*(1 + D[y[x],x]^2)^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1 \\
y(x)\to x \left (\frac {a^{2/3}}{x^{2/3}}-1\right )^{3/2}+c_1 \\
y(x)\to c_1-x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2} \\
y(x)\to x \left (-1-\frac {i \left (\sqrt {3}-i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1 \\
y(x)\to c_1-x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2} \\
y(x)\to x \left (-1+\frac {i \left (\sqrt {3}+i\right ) a^{2/3}}{2 x^{2/3}}\right )^{3/2}+c_1 \\
\end{align*}