60.1.506 problem 519
Internal
problem
ID
[10520]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
519
Date
solved
:
Sunday, March 30, 2025 at 05:43:00 PM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} {y^{\prime }}^{3}-f \left (x \right ) \left (a y^{2}+b y+c \right )^{2}&=0 \end{align*}
✓ Maple. Time used: 0.141 (sec). Leaf size: 195
ode:=diff(y(x),x)^3-f(x)*(a*y(x)^2+b*y(x)+c)^2 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
\int _{}^{y}\frac {1}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{{2}/{3}}}d \textit {\_a} -\frac {\int _{}^{x}{\left (f \left (\textit {\_a} \right ) \left (a y^{2}+b y+c \right )^{2}\right )}^{{1}/{3}}d \textit {\_a}}{\left (a y^{2}+b y+c \right )^{{2}/{3}}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{{2}/{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \int _{}^{x}{\left (f \left (\textit {\_a} \right ) \left (a y^{2}+b y+c \right )^{2}\right )}^{{1}/{3}}d \textit {\_a}}{2 \left (a y^{2}+b y+c \right )^{{2}/{3}}}+c_1 &= 0 \\
\int _{}^{y}\frac {1}{\left (\textit {\_a}^{2} a +\textit {\_a} b +c \right )^{{2}/{3}}}d \textit {\_a} -\frac {\left (i \sqrt {3}-1\right ) \int _{}^{x}{\left (f \left (\textit {\_a} \right ) \left (a y^{2}+b y+c \right )^{2}\right )}^{{1}/{3}}d \textit {\_a}}{2 \left (a y^{2}+b y+c \right )^{{2}/{3}}}+c_1 &= 0 \\
\end{align*}
✓ Mathematica. Time used: 15.206 (sec). Leaf size: 405
ode=-(f[x]*(c + b*y[x] + a*y[x]^2)^2) + D[y[x],x]^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {$\#$1} a+b) \left (\frac {a (\text {$\#$1} (\text {$\#$1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {(b+2 a \text {$\#$1})^2}{b^2-4 a c}\right )}{a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\&\right ]\left [\int _1^x\sqrt [3]{f(K[1])}dK[1]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {$\#$1} a+b) \left (\frac {a (\text {$\#$1} (\text {$\#$1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {(b+2 a \text {$\#$1})^2}{b^2-4 a c}\right )}{a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\&\right ]\left [\int _1^x-\sqrt [3]{-1} \sqrt [3]{f(K[2])}dK[2]+c_1\right ] \\
y(x)\to \text {InverseFunction}\left [\frac {\sqrt [3]{2} (2 \text {$\#$1} a+b) \left (\frac {a (\text {$\#$1} (\text {$\#$1} a+b)+c)}{4 a c-b^2}\right )^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2}{3},\frac {3}{2},\frac {(b+2 a \text {$\#$1})^2}{b^2-4 a c}\right )}{a (\text {$\#$1} (\text {$\#$1} a+b)+c)^{2/3}}\&\right ]\left [\int _1^x(-1)^{2/3} \sqrt [3]{f(K[3])}dK[3]+c_1\right ] \\
y(x)\to -\frac {\sqrt {b^2-4 a c}+b}{2 a} \\
y(x)\to \frac {\sqrt {b^2-4 a c}-b}{2 a} \\
\end{align*}
✓ Sympy. Time used: 1.906 (sec). Leaf size: 102
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
c = symbols("c")
y = Function("y")
ode = Eq(-(a*y(x)**2 + b*y(x) + c)**2*f(x) + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (y^{2} a + y b + c\right )^{\frac {2}{3}}}\, dy = C_{1} + \int \sqrt [3]{f{\left (x \right )}}\, dx, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (y^{2} a + y b + c\right )^{\frac {2}{3}}}\, dy = C_{1} + \frac {i \left (\sqrt {3} + i\right ) \int \sqrt [3]{f{\left (x \right )}}\, dx}{2}, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\left (y^{2} a + y b + c\right )^{\frac {2}{3}}}\, dy = C_{1} - \frac {i \left (\sqrt {3} - i\right ) \int \sqrt [3]{f{\left (x \right )}}\, dx}{2}\right ]
\]