2.4.33 problem 29
Internal
problem
ID
[8348]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
4.0
Problem
number
:
29
Date
solved
:
Sunday, November 10, 2024 at 09:11:07 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _dAlembert]
Solve
\begin{align*} \left (y-2 x y^{\prime }\right )^{2}&={y^{\prime }}^{3} \end{align*}
Let \(p=y^{\prime }\) the ode becomes
\begin{align*} \left (-2 x p +y \right )^{2} = p^{3} \end{align*}
Solving for \(y\) from the above results in
\begin{align*}
\tag{1} y &= 2 x p +p^{{3}/{2}} \\
\tag{2} y &= 2 x p -p^{{3}/{2}} \\
\end{align*}
This has the form
\begin{align*} y=xf(p)+g(p)\tag {*} \end{align*}
Where \(f,g\) are functions of \(p=y'(x)\). Each of the above ode’s is dAlembert ode which is now
solved.
Solving ode 1A
Taking derivative of (*) w.r.t. \(x\) gives
\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}
Comparing the form \(y=x f + g\) to (1A) shows that
\begin{align*} f &= 2 p\\ g &= p^{{3}/{2}} \end{align*}
Hence (2) becomes
\begin{align*} -p = \left (2 x +\frac {3 \sqrt {p}}{2}\right ) p^{\prime }\left (x \right )\tag {2A} \end{align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives
\begin{align*} -p = 0 \end{align*}
Solving the above for \(p\) results in
\begin{align*} p_{1} &=0 \end{align*}
Substituting these in (1A) and keeping singular solution that verifies the ode gives
\begin{align*} y = 0 \end{align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in
\begin{align*} p^{\prime }\left (x \right ) = -\frac {p \left (x \right )}{2 x +\frac {3 \sqrt {p \left (x \right )}}{2}}\tag {3} \end{align*}
Inverting the above ode gives
\begin{align*} \frac {d}{d p}x \left (p \right ) = -\frac {2 x \left (p \right )+\frac {3 \sqrt {p}}{2}}{p}\tag {4} \end{align*}
This ODE is now solved for \(x \left (p \right )\). The integrating factor is
\begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p}d p}\\ \mu &= p^{2}\\ \mu &= p^{2}\tag {5} \end{align*}
Integrating gives
\begin{align*} x \left (p \right )&= \frac {1}{\mu } \left ( \int { \mu \left (-\frac {3}{2 \sqrt {p}}\right ) \,dp} + c_1\right )\\ &= \frac {1}{\mu } \left (\frac {-3 p^{{5}/{2}}+5 c_1}{5 p^{2}}+c_1\right ) \\ &= \frac {-3 p^{{5}/{2}}+5 c_1}{5 p^{2}}\tag {5} \end{align*}
Now we need to eliminate \(p\) between the above solution and (1A). The first method is to solve
for \(p\) from Eq. (1A) and substitute the result into Eq. (5). The Second method is to solve for \(p\)
from Eq. (5) and substitute the result into (1A).
Eliminating \(p\) from the following two equations
\begin{align*}
x &= \frac {-3 p^{{5}/{2}}+5 c_1}{5 p^{2}} \\
y &= 2 x p +p^{{3}/{2}} \\
\end{align*}
results in
\begin{align*}
p &= \operatorname {RootOf}\left (3 \textit {\_Z}^{5}+5 x \,\textit {\_Z}^{4}-5 c_1 \right )^{2} \\
\end{align*}
Substituting the above into Eq
(1A) and simplifying gives
\begin{align*}
y &= 2 x \operatorname {RootOf}\left (3 \textit {\_Z}^{5}+5 x \,\textit {\_Z}^{4}-5 c_1 \right )^{2}+{\left (\operatorname {RootOf}\left (3 \textit {\_Z}^{5}+5 x \,\textit {\_Z}^{4}-5 c_1 \right )^{2}\right )}^{{3}/{2}} \\
\end{align*}
Solving ode 2A
Taking derivative of (*) w.r.t. \(x\) gives
\begin{align*} p &= f+(x f'+g') \frac {dp}{dx}\\ p-f &= (x f'+g') \frac {dp}{dx}\tag {2} \end{align*}
Comparing the form \(y=x f + g\) to (1A) shows that
\begin{align*} f &= 2 p\\ g &= -p^{{3}/{2}} \end{align*}
Hence (2) becomes
\begin{align*} -p = \left (2 x -\frac {3 \sqrt {p}}{2}\right ) p^{\prime }\left (x \right )\tag {2A} \end{align*}
The singular solution is found by setting \(\frac {dp}{dx}=0\) in the above which gives
\begin{align*} -p = 0 \end{align*}
Solving the above for \(p\) results in
\begin{align*} p_{1} &=0 \end{align*}
Substituting these in (1A) and keeping singular solution that verifies the ode gives
\begin{align*} y = 0 \end{align*}
The general solution is found when \( \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}x}}\neq 0\). From eq. (2A). This results in
\begin{align*} p^{\prime }\left (x \right ) = -\frac {p \left (x \right )}{2 x -\frac {3 \sqrt {p \left (x \right )}}{2}}\tag {3} \end{align*}
Inverting the above ode gives
\begin{align*} \frac {d}{d p}x \left (p \right ) = -\frac {2 x \left (p \right )-\frac {3 \sqrt {p}}{2}}{p}\tag {4} \end{align*}
This ODE is now solved for \(x \left (p \right )\). The integrating factor is
\begin{align*} \mu &= {\mathrm e}^{\int \frac {2}{p}d p}\\ \mu &= p^{2}\\ \mu &= p^{2}\tag {5} \end{align*}
Integrating gives
\begin{align*} x \left (p \right )&= \frac {1}{\mu } \left ( \int { \mu \left (\frac {3}{2 \sqrt {p}}\right ) \,dp} + c_2\right )\\ &= \frac {1}{\mu } \left (\frac {3 p^{{5}/{2}}+5 c_2}{5 p^{2}}+c_2\right ) \\ &= \frac {3 p^{{5}/{2}}+5 c_2}{5 p^{2}}\tag {5} \end{align*}
Now we need to eliminate \(p\) between the above solution and (1A). The first method is to solve
for \(p\) from Eq. (1A) and substitute the result into Eq. (5). The Second method is to solve for \(p\)
from Eq. (5) and substitute the result into (1A).
Eliminating \(p\) from the following two equations
\begin{align*}
x &= \frac {3 p^{{5}/{2}}+5 c_2}{5 p^{2}} \\
y &= 2 x p -p^{{3}/{2}} \\
\end{align*}
results in
\begin{align*}
p &= \operatorname {RootOf}\left (3 \textit {\_Z}^{5}-5 x \,\textit {\_Z}^{4}+5 c_2 \right )^{2} \\
\end{align*}
Substituting the above into Eq
(1A) and simplifying gives
\begin{align*}
y &= 2 x \operatorname {RootOf}\left (3 \textit {\_Z}^{5}-5 x \,\textit {\_Z}^{4}+5 c_2 \right )^{2}-{\left (\operatorname {RootOf}\left (3 \textit {\_Z}^{5}-5 x \,\textit {\_Z}^{4}+5 c_2 \right )^{2}\right )}^{{3}/{2}} \\
\end{align*}
The solution
\[
y = 2 x \operatorname {RootOf}\left (3 \textit {\_Z}^{5}+5 x \,\textit {\_Z}^{4}-5 c_1 \right )^{2}+{\left (\operatorname {RootOf}\left (3 \textit {\_Z}^{5}+5 x \,\textit {\_Z}^{4}-5 c_1 \right )^{2}\right )}^{{3}/{2}}
\]
was found not to satisfy the ode or the IC. Hence it
is removed. The solution
\[
y = 2 x \operatorname {RootOf}\left (3 \textit {\_Z}^{5}-5 x \,\textit {\_Z}^{4}+5 c_2 \right )^{2}-{\left (\operatorname {RootOf}\left (3 \textit {\_Z}^{5}-5 x \,\textit {\_Z}^{4}+5 c_2 \right )^{2}\right )}^{{3}/{2}}
\]
was found not to satisfy the ode or the IC. Hence it is
removed.
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (y \left (x \right )-2 x \left (\frac {d}{d x}y \left (x \right )\right )\right )^{2}=\left (\frac {d}{d x}y \left (x \right )\right )^{3} \\ \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 (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{6}-\frac {6 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}+\frac {4 x^{2}}{3}, \frac {d}{d x}y \left (x \right )=-\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{12}+\frac {3 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}+\frac {4 x^{2}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{6}+\frac {6 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}\right )}{2}, \frac {d}{d x}y \left (x \right )=-\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{12}+\frac {3 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}+\frac {4 x^{2}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{6}+\frac {6 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}\right )}{2}\right ] \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{6}-\frac {6 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}+\frac {4 x^{2}}{3} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{12}+\frac {3 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}+\frac {4 x^{2}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{6}+\frac {6 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Solve the equation}\hspace {3pt} \frac {d}{d x}y \left (x \right )=-\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{12}+\frac {3 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}+\frac {4 x^{2}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}{6}+\frac {6 \left (\frac {4 x y \left (x \right )}{3}-\frac {16 x^{4}}{9}\right )}{\left (-576 x^{3} y \left (x \right )+108 y \left (x \right )^{2}+512 x^{6}+12 \sqrt {-96 x^{3} y \left (x \right )^{3}+81 y \left (x \right )^{4}}\right )^{{1}/{3}}}\right )}{2} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\mathit {workingODE} , \mathit {workingODE} , \mathit {workingODE}\right \} \end {array} \]
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 simple symmetries for implicit equations
Successful isolation of dy/dx: 3 solutions were found. Trying to solve each resulting ODE.
*** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying homogeneous types:
trying exact
Looking for potential symmetries
trying an equivalence to an Abel ODE
trying 1st order ODE linearizable_by_differentiation
-> Solving 1st order ODE of high degree, Lie methods, 1st trial
`, `-> Computing symmetries using: way = 2
`, `-> Computing symmetries using: way = 2
-> Solving 1st order ODE of high degree, 2nd attempt. Trying parametric methods
trying dAlembert
<- dAlembert successful
<- dAlembert successful`
Maple dsolve solution
Solving time : 0.181
(sec)
Leaf size : 73
dsolve((-2*diff(y(x),x)*x+y(x))^2 = diff(y(x),x)^3,
y(x),singsol=all)
\begin{align*}
y &= 0 \\
\left [x \left (\textit {\_T} \right ) &= \frac {3 \textit {\_T}^{{5}/{2}}+5 c_{1}}{5 \textit {\_T}^{2}}, y \left (\textit {\_T} \right ) = \frac {\textit {\_T}^{{5}/{2}}+10 c_{1}}{5 \textit {\_T}}\right ] \\
\left [x \left (\textit {\_T} \right ) &= \frac {-3 \textit {\_T}^{{5}/{2}}+5 c_{1}}{5 \textit {\_T}^{2}}, y \left (\textit {\_T} \right ) = \frac {-\textit {\_T}^{{5}/{2}}+10 c_{1}}{5 \textit {\_T}}\right ] \\
\end{align*}
Mathematica DSolve solution
Solving time : 0.0
(sec)
Leaf size : 0
DSolve[{(y[x]-2*x*D[y[x],x])^2== D[y[x],x]^3,{}},
y[x],x,IncludeSingularSolutions->True]
Timed out