2.1.50 problem 50
Internal
problem
ID
[8533]
Book
:
Second
order
enumerated
odes
Section
:
section
1
Problem
number
:
50
Date
solved
:
Sunday, November 10, 2024 at 04:00:54 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]
Solve
\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{3}&=0 \end{align*}
Solved as second order missing x ode
Time used: 0.470 (sec)
This is missing independent variable second order ode. Solved by reduction of order by using
substitution which makes the dependent variable \(y\) an independent variable. Using
\begin{align*} y' &= p \end{align*}
Then
\begin{align*} y'' &= \frac {dp}{dx}\\ &= \frac {dp}{dy}\frac {dy}{dx}\\ &= p \frac {dp}{dy} \end{align*}
Hence the ode becomes
\begin{align*} y p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+p \left (y \right )^{3} = 0 \end{align*}
Which is now solved as first order ode for \(p(y)\).
The ode \(p^{\prime } = -\frac {p^{2}}{y}\) is separable as it can be written as
\begin{align*} p^{\prime }&= -\frac {p^{2}}{y}\\ &= f(y) g(p) \end{align*}
Where
\begin{align*} f(y) &= -\frac {1}{y}\\ g(p) &= p^{2} \end{align*}
Integrating gives
\begin{align*} \int { \frac {1}{g(p)} \,dp} &= \int { f(y) \,dy}\\ \int { \frac {1}{p^{2}}\,dp} &= \int { -\frac {1}{y} \,dy}\\ -\frac {1}{p}&=\ln \left (\frac {1}{y}\right )+c_1 \end{align*}
We now need to find the singular solutions, these are found by finding for what values \(g(p)\) is
zero, since we had to divide by this above. Solving \(g(p)=0\) or \(p^{2}=0\) for \(p\) gives
\begin{align*} p&=0 \end{align*}
Now we go over each such singular solution and check if it verifies the ode itself and
any initial conditions given. If it does not then the singular solution will not be
used.
Therefore the solutions found are
\begin{align*} -\frac {1}{p} = \ln \left (\frac {1}{y}\right )+c_1\\ p = 0 \end{align*}
Solving for \(p\) gives
\begin{align*}
p &= -\frac {1}{\ln \left (\frac {1}{y}\right )+c_1} \\
\end{align*}
For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order
ode to solve which is
\begin{align*} y^{\prime } = 0 \end{align*}
Since the ode has the form \(y^{\prime }=f(x)\), then we only need to integrate \(f(x)\).
\begin{align*} \int {dy} &= \int {0\, dx} + c_2 \\ y &= c_2 \end{align*}
For solution (2) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is
\begin{align*} y^{\prime } = -\frac {1}{\ln \left (\frac {1}{y}\right )+c_1} \end{align*}
Integrating gives
\begin{align*} \int \left (-\ln \left (\frac {1}{y}\right )-c_1 \right )d y &= dx\\ -y \left (\ln \left (\frac {1}{y}\right )+c_1 +1\right )&= x +c_3 \end{align*}
Solving for \(y\) gives
\begin{align*}
y &= \frac {x +c_3}{\operatorname {LambertW}\left (\left (x +c_3 \right ) {\mathrm e}^{-c_1 -1}\right )} \\
\end{align*}
Will add steps showing solving for IC soon.
Summary of solutions found
\begin{align*}
y &= c_2 \\
y &= \frac {x +c_3}{\operatorname {LambertW}\left (\left (x +c_3 \right ) {\mathrm e}^{-c_1 -1}\right )} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) y \left (x \right )+\left (\frac {d}{d x}y \left (x \right )\right )^{3}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Define new dependent variable}\hspace {3pt} u \\ {} & {} & u \left (x \right )=\frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ {} & {} & \frac {d}{d x}u \left (x \right )=\frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Use chain rule on the lhs}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y \left (x \right )\right ) \left (\frac {d}{d y}u \left (y \right )\right )=\frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Substitute in the definition of}\hspace {3pt} u \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )=\frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Make substitutions}\hspace {3pt} \frac {d}{d x}y \left (x \right )=u \left (y \right ),\frac {d^{2}}{d x^{2}}y \left (x \right )=u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right )\hspace {3pt}\textrm {to reduce order of ODE}\hspace {3pt} \\ {} & {} & u \left (y \right ) \left (\frac {d}{d y}u \left (y \right )\right ) y +u \left (y \right )^{3}=0 \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d y}u \left (y \right )=-\frac {u \left (y \right )^{2}}{y} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d y}u \left (y \right )}{u \left (y \right )^{2}}=-\frac {1}{y} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} y \\ {} & {} & \int \frac {\frac {d}{d y}u \left (y \right )}{u \left (y \right )^{2}}d y =\int -\frac {1}{y}d y +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\frac {1}{u \left (y \right )}=-\ln \left (y \right )+\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {1}{\ln \left (y \right )-\mathit {C1}} \\ \bullet & {} & \textrm {Solve 1st ODE for}\hspace {3pt} u \left (y \right ) \\ {} & {} & u \left (y \right )=\frac {1}{\ln \left (y \right )-\mathit {C1}} \\ \bullet & {} & \textrm {Revert to original variables with substitution}\hspace {3pt} u \left (y \right )=\frac {d}{d x}y \left (x \right ),y =y \left (x \right ) \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1}{\ln \left (y \left (x \right )\right )-\mathit {C1}} \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {1}{\ln \left (y \left (x \right )\right )-\mathit {C1}} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y \left (x \right )\right ) \left (\ln \left (y \left (x \right )\right )-\mathit {C1} \right )=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}y \left (x \right )\right ) \left (\ln \left (y \left (x \right )\right )-\mathit {C1} \right )d x =\int 1d x +\mathit {C2} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & -\mathit {C1} y \left (x \right )+y \left (x \right ) \ln \left (y \left (x \right )\right )-y \left (x \right )=x +\mathit {C2} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \left (x \right ) \\ {} & {} & y \left (x \right )={\mathrm e}^{\mathit {LambertW}\left (\left (x +\mathit {C2} \right ) {\mathrm e}^{-\mathit {C1} -1}\right )+\mathit {C1} +1} \end {array} \]
Maple trace
`Methods for second order ODEs:
--- Trying classification methods ---
trying 2nd order Liouville
trying 2nd order WeierstrassP
trying 2nd order JacobiSN
differential order: 2; trying a linearization to 3rd order
trying 2nd order ODE linearizable_by_differentiation
trying 2nd order, 2 integrating factors of the form mu(x,y)
trying differential order: 2; missing variables
<- differential order: 2; canonical coordinates successful
<- differential order 2; missing variables successful`
Maple dsolve solution
Solving time : 0.037
(sec)
Leaf size : 27
dsolve(y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^3 = 0,
y(x),singsol=all)
\begin{align*}
y &= 0 \\
y &= c_{1} \\
y &= \frac {x +c_{2}}{\operatorname {LambertW}\left (\left (x +c_{2} \right ) {\mathrm e}^{c_{1} -1}\right )} \\
\end{align*}
Mathematica DSolve solution
Solving time : 60.147
(sec)
Leaf size : 26
DSolve[{y[x]*D[y[x],{x,2}]+D[y[x],x]^3==0,{}},
y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {x+c_2}{W\left (e^{-1-c_1} (x+c_2)\right )}
\]