83.45.8 problem Ex 8 page 55
Internal
problem
ID
[19551]
Book
:
A
Text
book
for
differentional
equations
for
postgraduate
students
by
Ray
and
Chaturvedi.
First
edition,
1958.
BHASKAR
press.
INDIA
Section
:
Book
Solved
Excercises.
Chapter
IV.
Equations
of
the
first
order
but
not
of
the
first
degree
Problem
number
:
Ex
8
page
55
Date
solved
:
Tuesday, January 28, 2025 at 01:51:49 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3} x&=a +b y^{\prime } \end{align*}
✓ Solution by Maple
Time used: 0.024 (sec). Leaf size: 306
dsolve(x*diff(y(x),x)^3=a+b*diff(y(x),x),y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\int \frac {\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{2}/{3}}-b x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) 2^{{2}/{3}}}{{\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{1}/{3}} x}d x \right )}{12}+c_{1} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\int \frac {\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{2}/{3}}-\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) b x 2^{{2}/{3}}}{{\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{1}/{3}} x}d x \right )}{12}+c_{1} \\
y \left (x \right ) &= \frac {12^{{1}/{3}} \left (\int \frac {b 12^{{1}/{3}} x +{\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{2}/{3}}}{x {\left (\left (\sqrt {3}\, \sqrt {\frac {27 a^{2} x -4 b^{3}}{x}}+9 a \right ) x^{2}\right )}^{{1}/{3}}}d x \right )}{6}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 166.243 (sec). Leaf size: 535
DSolve[x*D[y[x],x]^3==a+b*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{6} x \left (-9 \sqrt {3} a x \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}+2\ 2^{2/3} \sqrt [3]{3} b \left (\sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-9 a x^2\right )^{4/3}+4 \sqrt [3]{2} b^2 x \left (3 \sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-27 a x^2\right )^{2/3}+81 a^2 x^3\right )}{\sqrt [3]{\sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-9 a x^2} \left (\sqrt [3]{2} \left (\sqrt {3} \sqrt {x^3 \left (27 a^2 x-4 b^3\right )}-9 a x^2\right )^{2/3}+2 \sqrt [3]{3} b x\right )^2}+c_1 \\
y(x)\to \int _1^x\frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) b K[1]+\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {K[1]^3 \left (27 a^2 K[1]-4 b^3\right )}-18 a K[1]^2\right )^{2/3}}{12 K[1] \sqrt [3]{\sqrt {3} \sqrt {K[1]^3 \left (27 a^2 K[1]-4 b^3\right )}-9 a K[1]^2}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) b K[2]+\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (2 \sqrt {3} \sqrt {K[2]^3 \left (27 a^2 K[2]-4 b^3\right )}-18 a K[2]^2\right )^{2/3}}{12 K[2] \sqrt [3]{\sqrt {3} \sqrt {K[2]^3 \left (27 a^2 K[2]-4 b^3\right )}-9 a K[2]^2}}dK[2]+c_1 \\
\end{align*}