29.34.20 problem 1022
Internal
problem
ID
[5591]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1022
Date
solved
:
Monday, January 27, 2025 at 12:13:00 PM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}+y^{\prime }+a -b x&=0 \end{align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 298
dsolve(diff(y(x),x)^3+diff(y(x),x)+a-b*x = 0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {\left (\int \frac {\left (i \sqrt {3}-1\right ) \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{2}/{3}}+12 i \sqrt {3}+12}{\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{1}/{3}}}d x \right )}{12}+c_{1} \\
y \left (x \right ) &= -\frac {\left (\int \frac {i \sqrt {3}\, \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{2}/{3}}+12 i \sqrt {3}+\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{2}/{3}}-12}{\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{1}/{3}}}d x \right )}{12}+c_{1} \\
y \left (x \right ) &= \frac {\left (\int \frac {\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{2}/{3}}-12}{\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 b x a +81 a^{2}+12}\right )^{{1}/{3}}}d x \right )}{6}+c_{1} \\
\end{align*}
✓ Solution by Mathematica
Time used: 2.555 (sec). Leaf size: 1086
DSolve[(D[y[x],x])^3 +D[y[x],x]+a-b x==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2^{2/3} \sqrt [3]{3} \left (27 a^2-54 a b x+27 b^2 x^2-2\right ) \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{2/3}-3\ 2^{2/3} 3^{5/6} (a-b x) \sqrt {27 a^2-54 a b x+27 b^2 x^2+4} \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{2/3}-4 \sqrt [3]{2} 3^{2/3} \left (27 a^2-54 a b x+27 b^2 x^2+1\right )+36 \sqrt [3]{2} \sqrt [6]{3} (a-b x) \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}+24 b c_1 \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{4/3}}{24 b \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{4/3}} \\
y(x)\to \frac {i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+i\right ) \left (27 a^2-54 a b x+27 b^2 x^2-2\right ) \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{2/3}+3\ 2^{2/3} 3^{5/6} \left (1-i \sqrt {3}\right ) (a-b x) \sqrt {27 a^2-54 a b x+27 b^2 x^2+4} \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{2/3}+4 \sqrt [3]{2} 3^{2/3} \left (1+i \sqrt {3}\right ) \left (27 a^2-54 a b x+27 b^2 x^2+1\right )-36 \sqrt [3]{2} \sqrt [6]{3} \left (1+i \sqrt {3}\right ) (a-b x) \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}+48 b c_1 \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{4/3}}{48 b \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{4/3}} \\
y(x)\to \frac {2^{2/3} \sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (-27 a^2+54 a b x-27 b^2 x^2+2\right ) \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{2/3}+3\ 2^{2/3} 3^{5/6} \left (1+i \sqrt {3}\right ) (a-b x) \sqrt {27 a^2-54 a b x+27 b^2 x^2+4} \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{2/3}+4 \sqrt [3]{2} 3^{2/3} \left (1-i \sqrt {3}\right ) \left (27 a^2-54 a b x+27 b^2 x^2+1\right )+36 i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) (a-b x) \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}+48 b c_1 \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{4/3}}{48 b \left (\sqrt {3} \sqrt {27 a^2-54 a b x+27 b^2 x^2+4}-9 a+9 b x\right )^{4/3}} \\
\end{align*}