53.2.8 problem 15
Internal
problem
ID
[8461]
Book
:
Elementary
differential
equations.
By
Earl
D.
Rainville,
Phillip
E.
Bedient.
Macmilliam
Publishing
Co.
NY.
6th
edition.
1981.
Section
:
CHAPTER
16.
Nonlinear
equations.
Section
97.
The
p-discriminant
equation.
EXERCISES
Page
314
Problem
number
:
15
Date
solved
:
Monday, January 27, 2025 at 04:02:22 PM
CAS
classification
:
[_dAlembert]
\begin{align*} {y^{\prime }}^{3}+x {y^{\prime }}^{2}-y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 984
dsolve(diff(y(x),x)^3+x*diff(y(x),x)^2-y(x)=0,y(x), singsol=all)
\begin{align*}
y &= 0 \\
y &= \frac {\left (4 x^{2}-2 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}+9\right )^{2} \left (4 x^{2}+4 x \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+12 x +3 \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}+9\right )}{-1728 x^{3}-7776 x^{2}-11664 x +23328 c_{1} +1296 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}+5832} \\
y &= \frac {\left (\frac {\left (-i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}}{4}+\left (2 x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}} \left (i-\sqrt {3}\right )}{4}-i \left (-x +\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (x +\frac {3}{2}\right )^{2} \left (\sqrt {3}+i\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\
y &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}}}{4}-\left (-2 x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (-i \sqrt {3}-1\right ) \left (x +\frac {3}{2}\right )^{2}\right ) {\left (\frac {\left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{2}/{3}} \left (\sqrt {3}+i\right )}{4}+i \left (x -\frac {3}{2}\right ) \left (-36 x^{2}-54 x +108 c_{1} -8 x^{3}+27+6 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}\right )^{{1}/{3}}+\left (x +\frac {3}{2}\right )^{2} \left (i-\sqrt {3}\right )\right )}^{2}}{216 x^{3}+972 x^{2}+1458 x -2916 c_{1} -162 \sqrt {-6 \left (1+2 c_{1} \right ) \left (4 x^{3}+18 x^{2}-27 c_{1} +27 x \right )}-729} \\
\end{align*}
✓ Solution by Mathematica
Time used: 80.947 (sec). Leaf size: 1489
DSolve[(D[y[x],x])^3+x*(D[y[x],x])^2-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {-16 x^4+8 \left (\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}-12\right ) x^3-4 \left (\left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}-9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right ) x^2+6 \left (72 c_1+2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+4 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}\right ) x+3 \left (4 c_1 \left (2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right )+9 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+12 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+2 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )} \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+27 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+81\right )}{24 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {(2 c_1+1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}} \\
y(x)\to -\frac {2 x^2}{3}+\frac {1}{6} i \left (\sqrt {3}+i\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2 x}{6 \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}+\frac {1}{96} \left (-\frac {i \left (\sqrt {3}-i\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}+i \left (\sqrt {3}+i\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}-4 x+6\right ){}^2+c_1 \\
y(x)\to -\frac {2 x^2}{3}-\frac {1}{6} i \left (\sqrt {3}-i\right ) x \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}+\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2 x}{6 \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}+\frac {1}{96} \left (\frac {i \left (\sqrt {3}+i\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}}-i \left (\sqrt {3}-i\right ) \sqrt [3]{-8 x^3-36 x^2+6 \sqrt {6} \sqrt {(1+2 c_1) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}-54 x+27+108 c_1}-4 x+6\right ){}^2+c_1 \\
y(x)\to 0 \\
\end{align*}