82.8.3 problem Ex. 3
Internal
problem
ID
[18751]
Book
:
Introductory
Course
On
Differential
Equations
by
Daniel
A
Murray.
Longmans
Green
and
Co.
NY.
1924
Section
:
Chapter
II.
Equations
of
the
first
order
and
of
the
first
degree.
Exercises
at
page
25
Problem
number
:
Ex.
3
Date
solved
:
Tuesday, January 28, 2025 at 12:14:26 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} 3 x^{2} y^{4}+2 y x +\left (2 x^{3} y^{3}-x^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.053 (sec). Leaf size: 271
dsolve((3*x^2*y(x)^4+2*x*y(x))+(2*x^3*y(x)^3-x^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {12^{{1}/{3}} \left (c_{1} 12^{{1}/{3}} x +\left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_{1}^{3}}{x}}\right )^{{2}/{3}}\right )}{6 x^{2} \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_{1}^{3}}{x}}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-1-i \sqrt {3}\right ) \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_{1}^{3}}{x}}\right )^{{2}/{3}}+\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_{1} 2^{{2}/{3}} x \right )}{12 \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_{1}^{3}}{x}}\right )^{{1}/{3}} x^{2}} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (1-i \sqrt {3}\right ) \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_{1}^{3}}{x}}\right )^{{2}/{3}}+c_{1} 2^{{2}/{3}} \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) x \right )}{12 \left (-9 x^{5}+x^{2} \sqrt {\frac {81 x^{7}-12 c_{1}^{3}}{x}}\right )^{{1}/{3}} x^{2}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.262 (sec). Leaf size: 349
DSolve[(3*x^2*y[x]^4+2*x*y[x])+(2*x^3*y[x]^3-x^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2 \sqrt [3]{3} e^{\frac {7 c_1}{3}} x^3+\sqrt [3]{2} \left (-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}\right ){}^{2/3}}{6^{2/3} x^3 \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}}{2 \sqrt [3]{2} 3^{2/3} x^3}-\frac {\left (\sqrt {3}+3 i\right ) e^{\frac {7 c_1}{3}}}{2^{2/3} 3^{5/6} \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}} \\
y(x)\to \frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}}{2 \sqrt [3]{2} 3^{2/3} x^3}-\frac {\left (\sqrt {3}-3 i\right ) e^{\frac {7 c_1}{3}}}{2^{2/3} 3^{5/6} \sqrt [3]{-9 x^8+\sqrt {81 x^{16}-12 e^{7 c_1} x^9}}} \\
\end{align*}