31.6.10 problem 10
Internal
problem
ID
[5759]
Book
:
Differential
Equations,
By
George
Boole
F.R.S.
1865
Section
:
Chapter
7
Problem
number
:
10
Date
solved
:
Monday, January 27, 2025 at 01:12:55 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: 0.040 (sec). Leaf size: 585
dsolve(x^2*(1+(diff(y(x),x))^2)^3-a^2=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {-\sqrt {\frac {x \left (x \,a^{2}\right )^{{1}/{3}} \left (a^{2}-\left (x \,a^{2}\right )^{{2}/{3}}\right )}{a^{2}}}\, a^{2}+c_{1} \left (x \,a^{2}\right )^{{2}/{3}}+\left (x \,a^{2}\right )^{{2}/{3}} \sqrt {\frac {x \left (x \,a^{2}\right )^{{1}/{3}} \left (a^{2}-\left (x \,a^{2}\right )^{{2}/{3}}\right )}{a^{2}}}}{\left (x \,a^{2}\right )^{{2}/{3}}} \\
y \left (x \right ) &= \frac {\left (a^{2}-\left (x \,a^{2}\right )^{{2}/{3}}\right ) \sqrt {\frac {x \left (x \,a^{2}\right )^{{1}/{3}} \left (a^{2}-\left (x \,a^{2}\right )^{{2}/{3}}\right )}{a^{2}}}+c_{1} \left (x \,a^{2}\right )^{{2}/{3}}}{\left (x \,a^{2}\right )^{{2}/{3}}} \\
y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (x \,a^{2}\right )^{{1}/{3}}+\left (x \,a^{2}\right )^{{1}/{3}}+2 x \right )}\, \sqrt {\frac {x \left (x \,a^{2}\right )^{{1}/{3}} \left (\sqrt {3}\, a^{2}-2 i \left (x \,a^{2}\right )^{{2}/{3}}-i a^{2}\right )}{a^{2}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (x \,a^{2}\right )^{{2}/{3}}+a^{2}\right )}{4 \sqrt {\left (\sqrt {3}\, \left (x \,a^{2}\right )^{{1}/{3}}-i \left (x \,a^{2}\right )^{{1}/{3}}-2 i x \right ) x}\, \left (x \,a^{2}\right )^{{2}/{3}}}+c_{1} \\
y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-x \left (i \sqrt {3}\, \left (x \,a^{2}\right )^{{1}/{3}}+\left (x \,a^{2}\right )^{{1}/{3}}+2 x \right )}\, \sqrt {\frac {x \left (x \,a^{2}\right )^{{1}/{3}} \left (\sqrt {3}\, a^{2}-2 i \left (x \,a^{2}\right )^{{2}/{3}}-i a^{2}\right )}{a^{2}}}\, \left (i \sqrt {3}\, a^{2}+2 \left (x \,a^{2}\right )^{{2}/{3}}+a^{2}\right )}{4 \sqrt {\left (\sqrt {3}\, \left (x \,a^{2}\right )^{{1}/{3}}-i \left (x \,a^{2}\right )^{{1}/{3}}-2 i x \right ) x}\, \left (x \,a^{2}\right )^{{2}/{3}}}+c_{1} \\
y \left (x \right ) &= \frac {\left (-2 \left (x \,a^{2}\right )^{{2}/{3}} \sqrt {2}+\left (i \sqrt {6}-\sqrt {2}\right ) a^{2}\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (x \,a^{2}\right )^{{2}/{3}}\right ) x \left (x \,a^{2}\right )^{{1}/{3}}}{a^{2}}}+4 c_{1} \left (x \,a^{2}\right )^{{2}/{3}}}{4 \left (x \,a^{2}\right )^{{2}/{3}}} \\
y \left (x \right ) &= -\frac {\left (-2 \left (x \,a^{2}\right )^{{2}/{3}} \sqrt {2}+\left (i \sqrt {6}-\sqrt {2}\right ) a^{2}\right ) \sqrt {\frac {\left (\left (i \sqrt {3}-1\right ) a^{2}-2 \left (x \,a^{2}\right )^{{2}/{3}}\right ) x \left (x \,a^{2}\right )^{{1}/{3}}}{a^{2}}}-4 c_{1} \left (x \,a^{2}\right )^{{2}/{3}}}{4 \left (x \,a^{2}\right )^{{2}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.327 (sec). Leaf size: 216
DSolve[x^2*(1+(D[y[x],x])^2)^3-a^2==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*}