83.23.10 problem 10
Internal
problem
ID
[19291]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Chapter
V.
Singular
solutions.
Exercise
V
at
page
76
Problem
number
:
10
Date
solved
:
Tuesday, January 28, 2025 at 01:24:33 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{2} \left (2-3 y\right )^{2}&=4-4 y \end{align*}
✓ Solution by Maple
Time used: 0.052 (sec). Leaf size: 469
dsolve(diff(y(x),x)^2*(2-3*y(x))^2=4*(1-y(x)),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= 1 \\
y \left (x \right ) &= -\frac {{\left (\left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}+\frac {12}{\left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}}\right )}^{2}}{36}+1 \\
y \left (x \right ) &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y \left (x \right ) &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (-108 x +108 c_{1} +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{2}/{3}}} \\
y \left (x \right ) &= -\frac {{\left (\left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}+\frac {12}{\left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{{1}/{3}}}\right )}^{2}}{36}+1 \\
y \left (x \right ) &= 1+\frac {{\left (\left (i-\sqrt {3}\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+12 \sqrt {3}\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 \left (x \right ) &= 1+\frac {{\left (\left (\sqrt {3}+i\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-12 \sqrt {3}\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*}
✓ Solution by Mathematica
Time used: 4.482 (sec). Leaf size: 896
DSolve[D[y[x],x]^2*(2-3*y[x])^2==4*(1-y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {1}{12} \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}+\frac {8}{\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}}+4\right ) \\
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}}+8\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}}+8\right ) \\
y(x)\to \frac {1}{12} \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}+\frac {8}{\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}}+4\right ) \\
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}}+8\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}}+8\right ) \\
y(x)\to 1 \\
\end{align*}