2.1.68 problem 68
Internal
problem
ID
[8206]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
1.0
Problem
number
:
68
Date
solved
:
Sunday, November 10, 2024 at 03:21:57 AM
CAS
classification
:
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve
\begin{align*} a y y^{\prime \prime }+b y&=c \end{align*}
Solved as second order missing x ode
Time used: 1.987 (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*} a y p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+b y = c \end{align*}
Which is now solved as first order ode for \(p(y)\).
The ode \(p^{\prime } = -\frac {b y -c}{p a y}\) is separable as it can be written as
\begin{align*} p^{\prime }&= -\frac {b y -c}{p a y}\\ &= f(y) g(p) \end{align*}
Where
\begin{align*} f(y) &= -\frac {b y -c}{a 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 {b y -c}{a y} \,dy}\\ \frac {p^{2}}{2}&=\frac {c \ln \left (y \right )-b y}{a}+c_1 \end{align*}
Solving for \(p\) gives
\begin{align*}
p &= \frac {\sqrt {2}\, \sqrt {a \left (c \ln \left (y \right )+c_1 a -b y \right )}}{a} \\
p &= -\frac {\sqrt {2}\, \sqrt {a \left (c \ln \left (y \right )+c_1 a -b y \right )}}{a} \\
\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 {2}\, \sqrt {a \left (c \ln \left (y\right )+c_1 a -b y\right )}}{a} \end{align*}
Unable to integrate (or intergal too complicated), and since no initial conditions are given,
then the result can be written as
\[ \int _{}^{y}\frac {a \sqrt {2}}{2 \sqrt {a \left (c \ln \left (\tau \right )+c_1 a -b \tau \right )}}d \tau = x +c_2 \]
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 {2}\, \sqrt {a \left (c \ln \left (y\right )+c_1 a -b y\right )}}{a} \end{align*}
Unable to integrate (or intergal too complicated), and since no initial conditions are
given, then the result can be written as
\[ \int _{}^{y}-\frac {a \sqrt {2}}{2 \sqrt {a \left (c \ln \left (\tau \right )+c_1 a -b \tau \right )}}d \tau = x +c_3 \]
Will add steps showing solving for IC
soon.
Summary of solutions found
\begin{align*}
\int _{}^{y}\frac {a \sqrt {2}}{2 \sqrt {a \left (c \ln \left (\tau \right )+c_1 a -b \tau \right )}}d \tau &= x +c_2 \\
\int _{}^{y}-\frac {a \sqrt {2}}{2 \sqrt {a \left (c \ln \left (\tau \right )+c_1 a -b \tau \right )}}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)+(_a*b-c)/(_a*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 : 0.056
(sec)
Leaf size : 68
dsolve(a*y(x)*diff(diff(y(x),x),x)+b*y(x) = c,
y(x),singsol=all)
\begin{align*}
a \left (\int _{}^{y}\frac {1}{\sqrt {a \left (2 \ln \left (\textit {\_a} \right ) c +c_{1} a -2 b \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
-a \left (\int _{}^{y}\frac {1}{\sqrt {a \left (2 \ln \left (\textit {\_a} \right ) c +c_{1} a -2 b \textit {\_a} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\
\end{align*}
Mathematica DSolve solution
Solving time : 0.47
(sec)
Leaf size : 43
DSolve[{a*y[x]*D[y[x],{x,2}]+b*y[x]==c,{}},
y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{y(x)}\frac {1}{\sqrt {c_1+\frac {2 (c \log (K[1])-b K[1])}{a}}}dK[1]{}^2=(x+c_2){}^2,y(x)\right ]
\]