1.559 problem 562
Internal
problem
ID
[9541]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
562
Date
solved
:
Thursday, October 17, 2024 at 08:32:40 PM
CAS
classification
:
[_dAlembert]
Solve
\begin{align*} a \left ({y^{\prime }}^{3}+1\right )^{{1}/{3}}+b x y^{\prime }-y&=0 \end{align*}
Solving for the derivative gives these ODE’s to solve
\begin{align*}
\tag{1} y^{\prime }&=\frac {a \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}{2 b^{3} x^{3}+2 a^{3}}-\frac {2 y^{2} b x \,a^{2}}{\left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}+\frac {y b^{2} x^{2}}{b^{3} x^{3}+a^{3}} \\
\tag{2} y^{\prime }&=-\frac {a \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}{4 \left (b^{3} x^{3}+a^{3}\right )}+\frac {y^{2} b x \,a^{2}}{\left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}+\frac {y b^{2} x^{2}}{b^{3} x^{3}+a^{3}}+\frac {i \sqrt {3}\, \left (\frac {a \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}{2 b^{3} x^{3}+2 a^{3}}+\frac {2 y^{2} b x \,a^{2}}{\left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}\right )}{2} \\
\tag{3} y^{\prime }&=-\frac {a \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}{4 \left (b^{3} x^{3}+a^{3}\right )}+\frac {y^{2} b x \,a^{2}}{\left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}+\frac {y b^{2} x^{2}}{b^{3} x^{3}+a^{3}}-\frac {i \sqrt {3}\, \left (\frac {a \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}{2 b^{3} x^{3}+2 a^{3}}+\frac {2 y^{2} b x \,a^{2}}{\left (b^{3} x^{3}+a^{3}\right ) \left (-4 b^{6} x^{6}-4 y^{3} b^{3} x^{3}-8 a^{3} b^{3} x^{3}+4 \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\, b^{3} x^{3}+4 y^{3} a^{3}-4 a^{6}+4 a^{3} \sqrt {b^{6} x^{6}+2 y^{3} b^{3} x^{3}+2 a^{3} b^{3} x^{3}+y^{6}-2 y^{3} a^{3}+a^{6}}\right )^{{1}/{3}}}\right )}{2} \\
\end{align*}
Now each of the above is solved
separately.
Solving Eq. (1)
Solving Eq. (2)
Solving Eq. (3)
1.559.1 Maple step by step solution
1.559.2 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
trying dAlembert
<- dAlembert successful`
1.559.3 Maple dsolve solution
Solving time : 0.154 (sec)
Leaf size : 3313
dsolve(a*(diff(y(x),x)^3+1)^(1/3)+b*x*diff(y(x),x)-y(x) = 0,
y(x),singsol=all)
\begin{align*}
\text {Expression too large to display} \\
\text {Expression too large to display} \\
\text {Expression too large to display} \\
\end{align*}
1.559.4 Mathematica DSolve solution
Solving time : 0.083 (sec)
Leaf size : 84
DSolve[{-y[x] + b*x*D[y[x],x] + a*(1 + D[y[x],x]^3)^(1/3)==0,{}},
y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\left \{x=K[1]^{\frac {b}{1-b}} \left (\frac {a \int \frac {K[1]^{\frac {2 b-1}{b-1}}}{\left (K[1]^3+1\right )^{2/3}}dK[1]}{1-b}+c_1\right ),y(x)=a \sqrt [3]{K[1]^3+1}+b x K[1]\right \},\{K[1],y(x)\}\right ]
\]