15.19.21 problem 21
Internal
problem
ID
[3305]
Book
:
Differential
Equations
by
Alfred
L.
Nelson,
Karl
W.
Folley,
Max
Coral.
3rd
ed.
DC
heath.
Boston.
1964
Section
:
Exercise
37,
page
171
Problem
number
:
21
Date
solved
:
Monday, January 27, 2025 at 07:32:52 AM
CAS
classification
:
[_quadrature]
\begin{align*} y {y^{\prime }}^{2}+2 y^{\prime }+1&=0 \end{align*}
✓ Solution by Maple
Time used: 0.044 (sec). Leaf size: 53
dsolve(diff(y(x),x)^2*y(x)+2*diff(y(x),x)+1=0,y(x), singsol=all)
\begin{align*}
\frac {\left (2 y \left (x \right )-2\right ) \sqrt {1-y \left (x \right )}}{3}+x -c_{1} +y \left (x \right )-1 &= 0 \\
\frac {\left (-2 y \left (x \right )+2\right ) \sqrt {1-y \left (x \right )}}{3}+x -c_{1} +y \left (x \right )-1 &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 28.113 (sec). Leaf size: 1111
DSolve[D[y[x],x]^2*y[x]+2*D[y[x],x]+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{3} \left (-24 x^2-12 (1+4 c_1) x+8 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+3-24 c_1{}^2-12 c_1\right ){}^{2/3}+\sqrt [3]{-24 x^2-12 (1+4 c_1) x+8 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+3-24 c_1{}^2-12 c_1}-8\ 3^{2/3} x-5\ 3^{2/3}-8\ 3^{2/3} c_1}{4 \sqrt [3]{-24 x^2-12 (1+4 c_1) x+8 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+3-24 c_1{}^2-12 c_1}} \\
y(x)\to \frac {3^{2/3} \left (1+i \sqrt {3}\right ) (8 x+5+8 c_1)}{8 \sqrt [3]{-24 x^2-12 (1+4 c_1) x+8 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+3-24 c_1{}^2-12 c_1}}+\frac {1}{8} i \left (\sqrt {3}+i\right ) \sqrt [3]{-72 x^2-36 (1+4 c_1) x+24 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+9-72 c_1{}^2-36 c_1}+\frac {1}{4} \\
y(x)\to \frac {3^{2/3} \left (1-i \sqrt {3}\right ) (8 x+5+8 c_1)}{8 \sqrt [3]{-24 x^2-12 (1+4 c_1) x+8 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+3-24 c_1{}^2-12 c_1}}-\frac {1}{8} \left (1+i \sqrt {3}\right ) \sqrt [3]{-72 x^2-36 (1+4 c_1) x+24 \sqrt {3} \sqrt {(x+1+c_1){}^3 (3 x+2+3 c_1)}+9-72 c_1{}^2-36 c_1}+\frac {1}{4} \\
y(x)\to \frac {\sqrt [3]{3} \left (-24 x^2-12 x+48 c_1 x+8 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+3-24 c_1{}^2+12 c_1\right ){}^{2/3}+\sqrt [3]{-24 x^2-12 x+48 c_1 x+8 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+3-24 c_1{}^2+12 c_1}-8\ 3^{2/3} x-5\ 3^{2/3}+8\ 3^{2/3} c_1}{4 \sqrt [3]{-24 x^2+12 (-1+4 c_1) x+8 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+3-24 c_1{}^2+12 c_1}} \\
y(x)\to \frac {3^{2/3} \left (1+i \sqrt {3}\right ) (8 x+5-8 c_1)}{8 \sqrt [3]{-24 x^2+12 (-1+4 c_1) x+8 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+3-24 c_1{}^2+12 c_1}}+\frac {1}{8} i \left (\sqrt {3}+i\right ) \sqrt [3]{-72 x^2+36 (-1+4 c_1) x+24 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+9-72 c_1{}^2+36 c_1}+\frac {1}{4} \\
y(x)\to \frac {3^{2/3} \left (1-i \sqrt {3}\right ) (8 x+5-8 c_1)}{8 \sqrt [3]{-24 x^2+12 (-1+4 c_1) x+8 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+3-24 c_1{}^2+12 c_1}}-\frac {1}{8} \left (1+i \sqrt {3}\right ) \sqrt [3]{-72 x^2+36 (-1+4 c_1) x+24 \sqrt {3} \sqrt {(3 x+2-3 c_1) (x+1-c_1){}^3}+9-72 c_1{}^2+36 c_1}+\frac {1}{4} \\
\end{align*}