2.1.67 problem 67
Internal
problem
ID
[8205]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
67
Date
solved
:
Sunday, November 10, 2024 at 03:09:44 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve
\begin{align*} 3 y y^{\prime \prime }+y&=5 \end{align*}
Solved as second order missing x ode
Time used: 732.357 (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*} 3 y p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+y = 5 \end{align*}
Which is now solved as first order ode for \(p(y)\).
The ode \(p^{\prime } = -\frac {y -5}{3 p y}\) is separable as it can be written as
\begin{align*} p^{\prime }&= -\frac {y -5}{3 p y}\\ &= f(y) g(p) \end{align*}
Where
\begin{align*} f(y) &= -\frac {y -5}{3 y}\\ g(p) &= \frac {1}{p} \end{align*}
Integrating gives
\begin{align*} \int { \frac {1}{g(p)} \,dp} &= \int { f(y) \,dy}\\ \int { p\,dp} &= \int { -\frac {y -5}{3 y} \,dy}\\ \frac {p^{2}}{2}&=-\frac {y}{3}+\ln \left (y^{{5}/{3}}\right )+c_1 \end{align*}
Solving for \(p\) gives
\begin{align*}
p &= -\frac {\sqrt {18 \ln \left (y^{{5}/{3}}\right )+18 c_1 -6 y}}{3} \\
p &= \frac {\sqrt {18 \ln \left (y^{{5}/{3}}\right )+18 c_1 -6 y}}{3} \\
\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 } = -\frac {\sqrt {18 \ln \left (y^{{5}/{3}}\right )+18 c_1 -6 y}}{3} \end{align*}
Unable to integrate (or intergal too complicated), and since no initial conditions are
given, then the result can be written as
\[ \int _{}^{y}-\frac {3}{\sqrt {18 \ln \left (\tau ^{{5}/{3}}\right )+18 c_1 -6 \tau }}d \tau = x +c_2 \]
Singular solutions are found by solving
\begin{align*} -\frac {\sqrt {18 \ln \left (y^{{5}/{3}}\right )+18 c_1 -6 y}}{3}&= 0 \end{align*}
for \(y\). This is because we had to divide by this in the above step. This gives the following
singular solution(s), which also have to satisfy the given ODE.
\begin{align*} y = \left ({\mathrm e}^{-\frac {5 \operatorname {LambertW}\left (-\frac {243^{{4}/{5}} {\mathrm e}^{-\frac {3 c_1}{5}}}{405}\right )}{3}-c_1}\right )^{{3}/{5}} \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 {\sqrt {18 \ln \left (y^{{5}/{3}}\right )+18 c_1 -6 y}}{3} \end{align*}
Unable to integrate (or intergal too complicated), and since no initial conditions are
given, then the result can be written as
\[ \int _{}^{y}\frac {3}{\sqrt {18 \ln \left (\tau ^{{5}/{3}}\right )+18 c_1 -6 \tau }}d \tau = x +c_3 \]
Singular solutions are found by solving
\begin{align*} \frac {\sqrt {18 \ln \left (y^{{5}/{3}}\right )+18 c_1 -6 y}}{3}&= 0 \end{align*}
for \(y\). This is because we had to divide by this in the above step. This gives the following
singular solution(s), which also have to satisfy the given ODE.
\begin{align*} y = \left ({\mathrm e}^{-\frac {5 \operatorname {LambertW}\left (-\frac {243^{{4}/{5}} {\mathrm e}^{-\frac {3 c_1}{5}}}{405}\right )}{3}-c_1}\right )^{{3}/{5}} \end{align*}
Will add steps showing solving for IC soon.
The solution
\[
\int _{}^{y}-\frac {3}{\sqrt {18 \ln \left (\tau ^{{5}/{3}}\right )+18 c_1 -6 \tau }}d \tau = x +c_2
\]
was found not to satisfy the ode or the IC. Hence it is removed. The solution
\[
y = \left ({\mathrm e}^{-\frac {5 \operatorname {LambertW}\left (-\frac {243^{{4}/{5}} {\mathrm e}^{-\frac {3 c_1}{5}}}{405}\right )}{3}-c_1}\right )^{{3}/{5}}
\]
was found not to satisfy the ode or the IC. Hence it is removed.
Summary of solutions found
\begin{align*}
\int _{}^{y}\frac {3}{\sqrt {18 \ln \left (\tau ^{{5}/{3}}\right )+18 c_1 -6 \tau }}d \tau &= x +c_3 \\
\end{align*}
Maple step by step solution
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
`, `-> Computing symmetries using: way = 3
`, `-> Computing symmetries using: way = exp_sym
-> Calling odsolve with the ODE`, (diff(_b(_a), _a))*_b(_a)+(1/3)*(_a-5)/_a = 0, _b(_a)` *** Sublevel 2 ***
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
<- Bernoulli successful
<- differential order: 2; canonical coordinates successful
<- differential order 2; missing variables successful`
Maple dsolve solution
Solving time : 0.062
(sec)
Leaf size : 59
dsolve(3*y(x)*diff(diff(y(x),x),x)+y(x) = 5,
y(x),singsol=all)
\begin{align*}
-3 \left (\int _{}^{y}\frac {1}{\sqrt {30 \ln \left (\textit {\_a} \right )+9 c_{1} -6 \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
3 \left (\int _{}^{y}\frac {1}{\sqrt {30 \ln \left (\textit {\_a} \right )+9 c_{1} -6 \textit {\_a}}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
Mathematica DSolve solution
Solving time : 0.324
(sec)
Leaf size : 41
DSolve[{3*y[x]*D[y[x],{x,2}]+y[x]==5,{}},
y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\frac {1}{\sqrt {c_1+\frac {2}{3} (5 \log (K[1])-K[1])}}dK[1]{}^2=(x+c_2){}^2,y(x)\right ]
\]