77.1.50 problem 69 (page 112)
Internal
problem
ID
[17940]
Book
:
V.V.
Stepanov,
A
course
of
differential
equations
(in
Russian),
GIFML.
Moscow
(1958)
Section
:
All
content
Problem
number
:
69
(page
112)
Date
solved
:
Tuesday, January 28, 2025 at 11:14:14 AM
CAS
classification
:
[_quadrature]
\begin{align*} x {y^{\prime }}^{3}&=1+y^{\prime } \end{align*}
✓ Solution by Maple
Time used: 0.128 (sec). Leaf size: 246
dsolve(x*diff(y(x),x)^3=1+diff(y(x),x),y(x), singsol=all)
\begin{align*}
y &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\int -\frac {\left (1-i \sqrt {3}\right ) \left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{2}/{3}}+x \left (3^{{1}/{3}}+i 3^{{5}/{6}}\right ) 2^{{2}/{3}}}{\left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{1}/{3}} x}d x \right )}{12}+c_{1} \\
y &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\int \frac {\left (1+i \sqrt {3}\right ) \left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{2}/{3}}+\left (-i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x 2^{{2}/{3}}}{\left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{1}/{3}} x}d x \right )}{12}+c_{1} \\
y &= \frac {12^{{1}/{3}} \left (\int \frac {12^{{1}/{3}} x +\left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{2}/{3}}}{x \left (\left (\sqrt {3}\, \sqrt {\frac {-4+27 x}{x}}+9\right ) x^{2}\right )^{{1}/{3}}}d x \right )}{6}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 174.180 (sec). Leaf size: 363
DSolve[x*D[y[x],x]^3==1+D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \int _1^x\frac {2 \sqrt [3]{3} K[1]+\sqrt [3]{2} \left (9 K[1]^2+\sqrt {3} \sqrt {K[1]^3 (27 K[1]-4)}\right )^{2/3}}{6^{2/3} K[1] \sqrt [3]{9 K[1]^2+\sqrt {3} \sqrt {K[1]^3 (27 K[1]-4)}}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (18 K[2]^2+2 \sqrt {3} \sqrt {K[2]^3 (27 K[2]-4)}\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) K[2]}{12 K[2] \sqrt [3]{9 K[2]^2+\sqrt {3} \sqrt {K[2]^3 (27 K[2]-4)}}}dK[2]+c_1 \\
y(x)\to \int _1^x-\frac {i \left (\sqrt [3]{2} \sqrt [6]{3} \left (-6-2 i \sqrt {3}\right ) K[3]+\sqrt [3]{3} \left (-i+\sqrt {3}\right ) \left (18 K[3]^2+2 \sqrt {3} \sqrt {K[3]^3 (27 K[3]-4)}\right )^{2/3}\right )}{12 K[3] \sqrt [3]{9 K[3]^2+\sqrt {3} \sqrt {K[3]^3 (27 K[3]-4)}}}dK[3]+c_1 \\
\end{align*}