29.35.13 problem 1046
Internal
problem
ID
[5611]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
35
Problem
number
:
1046
Date
solved
:
Monday, January 27, 2025 at 12:26:43 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}+\left (1-3 x \right ) {y^{\prime }}^{2}-x \left (1-3 x \right ) y^{\prime }-1-x^{3}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.115 (sec). Leaf size: 376
dsolve(diff(y(x),x)^3+(1-3*x)*diff(y(x),x)^2-x*(1-3*x)*diff(y(x),x)-1-x^3 = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (\int \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}+12 \left (i \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}+1\right ) \left (x -\frac {1}{3}\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}d x \right )}{12}+c_{1} \\
y \left (x \right ) &= -\frac {\left (\int \frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}+12 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}-1\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}d x \right )}{12}+c_{1} \\
y \left (x \right ) &= \frac {\left (\int \frac {4+6 \left (-2+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}\right ) x +\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{2}/{3}}-2 \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{{1}/{3}}}d x \right )}{6}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 94.042 (sec). Leaf size: 1649
DSolve[(D[y[x],x])^3+(1-3*x)(D[y[x],x])^2-x*(1-3*x)*D[y[x],x]-1 -x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {3 \sqrt {12 x^3-3 x^2+54 x+69} \left (12 \sqrt [3]{2} (6 x-1) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-112\ 2^{2/3} \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}-48 (1-3 x)^2+36 x+324\right )-36 \sqrt [3]{2} \left (21 x^2+45 x-8\right ) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+112\ 2^{2/3} (9 x+25) \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}+120 (3 x-1)^3-1344 (3 x-1)+1308 (1-3 x)^2-9408}{324\ 2^{2/3} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{4/3}}+\frac {1}{3} \log \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )-\log \left (\sqrt [3]{2} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+2^{2/3} \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}-6 x+2\right )+\frac {1}{6} x (3 x-2)+c_1 \\
y(x)\to \frac {3 \sqrt {12 x^3-3 x^2+54 x+69} \left (24 \sqrt [3]{2} (6 x-1) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+448\ 2^{2/3} \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}-96 (1-3 x)^2+72 x+648\right )+24 i \sqrt {3} \left (-3 \sqrt {12 x^3-3 x^2+54 x+69} \left (\sqrt [3]{2} (1-6 x) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-4 (1-3 x)^2+3 x+27\right )-3 \sqrt [3]{2} \left (21 x^2+45 x-8\right ) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-10 (3 x-1)^3+112 (3 x-1)-109 (1-3 x)^2+784\right )-72 \sqrt [3]{2} \left (21 x^2+45 x-8\right ) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-448\ 2^{2/3} (9 x+25) \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}+240 (3 x-1)^3-2688 (3 x-1)+2616 (1-3 x)^2-18816}{1296\ 2^{2/3} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{4/3}}+\frac {1}{3} \log \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )-\log \left (-i \sqrt [3]{2} \sqrt {3} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-\sqrt [3]{2} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+2\ 2^{2/3} \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}+\left (6-6 i \sqrt {3}\right ) x+2 i \sqrt {3}-2\right )+\frac {1}{6} x (3 x-2)+c_1 \\
y(x)\to \frac {3 \sqrt {12 x^3-3 x^2+54 x+69} \left (24 \sqrt [3]{2} (6 x-1) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+448\ 2^{2/3} \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}-96 (1-3 x)^2+72 x+648\right )+\sqrt {3} \left (72 i \sqrt {12 x^3-3 x^2+54 x+69} \left (\sqrt [3]{2} (1-6 x) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-4 (1-3 x)^2+3 x+27\right )+72 i \sqrt [3]{2} \left (21 x^2+45 x-8\right ) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+240 i (3 x-1)^3-2688 i (3 x-1)+2616 i (1-3 x)^2-18816 i\right )-72 \sqrt [3]{2} \left (21 x^2+45 x-8\right ) \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-448\ 2^{2/3} (9 x+25) \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}+240 (3 x-1)^3-2688 (3 x-1)+2616 (1-3 x)^2-18816}{1296\ 2^{2/3} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{4/3}}+\frac {1}{3} \log \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )-\log \left (i \sqrt [3]{2} \sqrt {3} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}-\sqrt [3]{2} \left (3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25\right )^{2/3}+2\ 2^{2/3} \sqrt [3]{3 \sqrt {12 x^3-3 x^2+54 x+69}-9 x-25}+\left (6+6 i \sqrt {3}\right ) x-2 i \sqrt {3}-2\right )+\frac {1}{6} x (3 x-2)+c_1 \\
\end{align*}