1.520 problem 523
Internal
problem
ID
[9502]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
523
Date
solved
:
Thursday, October 17, 2024 at 03:50:37 PM
CAS
classification
:
[_quadrature]
Solve
\begin{align*} {y^{\prime }}^{3}-a x y^{\prime }+x^{3}&=0 \end{align*}
Solving for the derivative gives these ODE’s to solve
\begin{align*}
\tag{1} y^{\prime }&=\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}} \\
\tag{2} y^{\prime }&=-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2} \\
\tag{3} y^{\prime }&=-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2} \\
\end{align*}
Now each of the above is solved
separately.
Solving Eq. (1)
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
\begin{align*} \int {dy} &= \int {\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}+12 a x}{6 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\, dx}\\ y &= \int \frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}+12 a x}{6 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}d x + c_1 \end{align*}
\begin{align*} y&= \int \frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}+12 a x}{6 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}d x +c_1 \end{align*}
Solving Eq. (2)
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
\begin{align*} \int {dy} &= \int {\frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 i \sqrt {3}\, a x -\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\, dx}\\ y &= \int \frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 i \sqrt {3}\, a x -\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}d x + c_2 \end{align*}
\begin{align*} y&= \int \frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 i \sqrt {3}\, a x -\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}d x +c_2 \end{align*}
Solving Eq. (3)
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
\begin{align*} \int {dy} &= \int {-\frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 i \sqrt {3}\, a x +\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}+12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\, dx}\\ y &= \int -\frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 i \sqrt {3}\, a x +\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}+12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}d x + c_3 \end{align*}
\begin{align*} y&= \int -\frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}-12 i \sqrt {3}\, a x +\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{2}/{3}}+12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}d x +c_3 \end{align*}
1.520.1 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y \left (x \right )\right )^{3}-a x \left (\frac {d}{d x}y \left (x \right )\right )+x^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d x}y \left (x \right )=\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}, \frac {d}{d x}y \left (x \right )=-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}, \frac {d}{d x}y \left (x \right )=-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\int \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )d x +\textit {\_C1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )=\int \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )d x +\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int \left (-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right )d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\int \left (-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right )d x +\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right )d x =\int \left (-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right )d x +\textit {\_C1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y \left (x \right )=\int \left (-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right )d x +\textit {\_C1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y \left (x \right )=\int \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}+\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )d x +\mathit {C1} , y \left (x \right )=\int \left (-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right )d x +\mathit {C1} , y \left (x \right )=\int \left (-\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{12}-\frac {a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}{6}-\frac {2 a x}{\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{{1}/{3}}}\right )}{2}\right )d x +\mathit {C1} \right \} \end {array} \]
1.520.2 Maple trace
`Methods for first order ODEs:
-> Solving 1st order ODE of high degree, 1st attempt
trying 1st order WeierstrassP solution for high degree ODE
trying 1st order WeierstrassPPrime solution for high degree ODE
trying 1st order JacobiSN solution for high degree ODE
trying 1st order ODE linearizable_by_differentiation
trying differential order: 1; missing variables
<- differential order: 1; missing y(x) successful`
1.520.3 Maple dsolve solution
Solving time : 0.014 (sec)
Leaf size : 248
dsolve(diff(y(x),x)^3-a*x*diff(y(x),x)+x^3 = 0,
y(x),singsol=all)
\begin{align*}
y &= -\frac {\left (\int \left (\left (1+i \sqrt {3}\right ) \left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}-\frac {12 a \left (i \sqrt {3}-1\right ) x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}}\right )d x \right )}{12}+c_1 \\
y &= \frac {\left (\int \left (\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )-\frac {12 \left (1+i \sqrt {3}\right ) a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}}\right )d x \right )}{12}+c_1 \\
y &= \frac {\left (\int \frac {\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{2}/{3}}+12 a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}}d x \right )}{6}+c_1 \\
\end{align*}
1.520.4 Mathematica DSolve solution
Solving time : 146.871 (sec)
Leaf size : 349
DSolve[{x^3 - a*x*D[y[x],x] + D[y[x],x]^3==0,{}},
y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \int _1^x\frac {2 \sqrt [3]{3} a K[1]+\sqrt [3]{2} \left (\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3\right )^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (2 \sqrt {81 K[2]^6-12 a^3 K[2]^3}-18 K[2]^3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) a K[2]}{12 \sqrt [3]{\sqrt {81 K[2]^6-12 a^3 K[2]^3}-9 K[2]^3}}dK[2]+c_1 \\
y(x)\to \int _1^x\frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {81 K[3]^6-12 a^3 K[3]^3}-18 K[3]^3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) a K[3]}{12 \sqrt [3]{\sqrt {81 K[3]^6-12 a^3 K[3]^3}-9 K[3]^3}}dK[3]+c_1 \\
\end{align*}