60.1.529 problem 532
Internal
problem
ID
[10543]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
532
Date
solved
:
Monday, January 27, 2025 at 08:53:21 PM
CAS
classification
:
[_quadrature]
\begin{align*} a {y^{\prime }}^{3}+b {y^{\prime }}^{2}+c y^{\prime }-y-d&=0 \end{align*}
✓ Solution by Maple
Time used: 0.271 (sec). Leaf size: 911
dsolve(a*diff(y(x),x)^3+b*diff(y(x),x)^2+c*diff(y(x),x)-y(x)-d=0,y(x), singsol=all)
\begin{align*}
3 \sqrt {3}\, 2^{{1}/{3}} a \left (\int _{}^{y}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{1}/{3}}}{\sqrt {3}\, 2^{{1}/{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{1}/{3}} b -3^{{1}/{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{2}/{3}}+3 \left (a c -\frac {b^{2}}{3}\right ) 2^{{2}/{3}} 3^{{2}/{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\
\frac {12 \sqrt {3}\, 2^{{1}/{3}} a \left (\int _{}^{y}\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{1}/{3}}}{-3 \left (i-\frac {\sqrt {3}}{3}\right ) b 2^{{1}/{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{1}/{3}}+2 \,3^{{1}/{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{2}/{3}}+9 \left (a c -\frac {b^{2}}{3}\right ) 2^{{2}/{3}} \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right )}d \textit {\_a} \right )+\left (1+i \sqrt {3}\right ) \left (x -c_{1} \right )}{1+i \sqrt {3}} &= 0 \\
\frac {12 i \sqrt {3}\, 2^{{1}/{3}} a \left (\int _{}^{y}-\frac {\left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{1}/{3}}}{-3 b 2^{{1}/{3}} \left (i+\frac {\sqrt {3}}{3}\right ) \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{1}/{3}}-2 \,3^{{1}/{3}} \left (9 \sqrt {27 \left (d +\textit {\_a} \right )^{2} a^{2}+18 \left (\left (d +\textit {\_a} \right ) b +\frac {2 c^{2}}{9}\right ) c a +\left (-4 \textit {\_a} -4 d \right ) b^{3}-b^{2} c^{2}}\, a +27 \sqrt {3}\, \left (a^{2} \left (d +\textit {\_a} \right )+\frac {a b c}{3}-\frac {2 b^{3}}{27}\right )\right )^{{2}/{3}}+9 \left (i 3^{{1}/{6}}-\frac {3^{{2}/{3}}}{3}\right ) \left (a c -\frac {b^{2}}{3}\right ) 2^{{2}/{3}}}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.469 (sec). Leaf size: 1064
DSolve[-d - y[x] + c*D[y[x],x] + b*D[y[x],x]^2 + a*D[y[x],x]^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}}}{2 \sqrt [3]{2} b^2+2 \sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}} b-6 \sqrt [3]{2} a c+2^{2/3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}}d\text {$\#$1}\&\right ]\left [-\frac {x}{6 a}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}}}{2 i \sqrt [3]{2} \sqrt {3} b^2+2 \sqrt [3]{2} b^2-4 \sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}} b-6 i \sqrt [3]{2} \sqrt {3} a c-6 \sqrt [3]{2} a c-i 2^{2/3} \sqrt {3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}+2^{2/3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}}d\text {$\#$1}\&\right ]\left [\frac {x}{12 a}+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\int \frac {\sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}}}{-2 i \sqrt [3]{2} \sqrt {3} b^2+2 \sqrt [3]{2} b^2-4 \sqrt [3]{2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}} b+6 i \sqrt [3]{2} \sqrt {3} a c-6 \sqrt [3]{2} a c+i 2^{2/3} \sqrt {3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}+2^{2/3} \left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}+\sqrt {4 \left (3 a c-b^2\right )^3+\left (2 b^3-9 a c b-27 a^2 d-27 a^2 \text {$\#$1}\right )^2}\right )^{2/3}}d\text {$\#$1}\&\right ]\left [\frac {x}{12 a}+c_1\right ] \\
y(x)\to -d \\
\end{align*}