34.6.5 problem 14
Internal
problem
ID
[7972]
Book
:
Schaums
Outline.
Theory
and
problems
of
Differential
Equations,
1st
edition.
Frank
Ayres.
McGraw
Hill
1952
Section
:
Chapter
10.
Singular
solutions,
Extraneous
loci.
Supplemetary
problems.
Page
74
Problem
number
:
14
Date
solved
:
Tuesday, September 30, 2025 at 05:12:32 PM
CAS
classification
:
[_quadrature]
\begin{align*} \left (3 y-1\right )^{2} {y^{\prime }}^{2}&=4 y \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 511
ode:=(3*y(x)-1)^2*diff(y(x),x)^2 = 4*y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= 0 \\
y &= \frac {{\left (\left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12\right )}^{2}}{36 \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y &= \frac {{\left (i \sqrt {3}\, \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}-12 i \sqrt {3}+\left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12\right )}^{2}}{144 \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y &= \frac {{\left (\left (i \sqrt {3}-1\right ) \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}-12 i \sqrt {3}-12\right )}^{2}}{144 \left (108 c_1 -108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y &= \frac {{\left (\left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}+12\right )}^{2}}{36 \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y &= \frac {{\left (i \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}} \sqrt {3}+\left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}-12 i \sqrt {3}+12\right )}^{2}}{144 \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y &= \frac {{\left (\left (i \sqrt {3}-1\right ) \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}-12 i \sqrt {3}-12\right )}^{2}}{144 \left (-108 c_1 +108 x +12 \sqrt {81 c_1^{2}-162 c_1 x +81 x^{2}-12}\right )^{{2}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 3.537 (sec). Leaf size: 892
ode=(3*y[x]-1)^2*D[y[x],x]^2==4*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\left (2+\sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x-8+27 c_1{}^2}\right ){}^2}{6 \sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x-8+27 c_1{}^2}}\\ y(x)&\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x-8+27 c_1{}^2}+\frac {-8-8 i \sqrt {3}}{\sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x-8+27 c_1{}^2}}+16\right )\\ y(x)&\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x-8+27 c_1{}^2}+\frac {-8+8 i \sqrt {3}}{\sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(-2 x+c_1){}^2 \left (108 x^2-108 c_1 x-16+27 c_1{}^2\right )}-108 c_1 x-8+27 c_1{}^2}}+16\right )\\ y(x)&\to \frac {\left (2+\sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x-8+27 c_1{}^2}\right ){}^2}{6 \sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x-8+27 c_1{}^2}}\\ y(x)&\to \frac {1}{24} \left (2 i \left (\sqrt {3}+i\right ) \sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x-8+27 c_1{}^2}-\frac {8 \left (1+i \sqrt {3}\right )}{\sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x-8+27 c_1{}^2}}+16\right )\\ y(x)&\to \frac {1}{24} \left (-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x-8+27 c_1{}^2}+\frac {8 i \left (\sqrt {3}+i\right )}{\sqrt [3]{108 x^2+3 \sqrt {3} \sqrt {(2 x+c_1){}^2 \left (108 x^2+108 c_1 x-16+27 c_1{}^2\right )}+108 c_1 x-8+27 c_1{}^2}}+16\right )\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((3*y(x) - 1)**2*Derivative(y(x), x)**2 - 4*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out