2.1.23 problem 23
Internal
problem
ID
[8433]
Book
:
First
order
enumerated
odes
Section
:
section
1
Problem
number
:
23
Date
solved
:
Sunday, November 10, 2024 at 03:46:03 AM
CAS
classification
:
[_rational, _Bernoulli]
Solve
\begin{align*} c y^{\prime }&=\frac {a x +b y^{2}}{y} \end{align*}
Solved as first order Bernoulli ode
Time used: 0.357 (sec)
In canonical form, the ODE is
\begin{align*} y' &= F(x,y)\\ &= \frac {b \,y^{2}+a x}{y c} \end{align*}
This is a Bernoulli ODE.
\[ y' = \left (\frac {b}{c}\right ) y + \left (\frac {a x}{c}\right )\frac {1}{y} \tag {1} \]
The standard Bernoulli ODE has the form
\[ y' = f_0(x)y+f_1(x)y^n \tag {2} \]
Comparing this to (1)
shows that
\begin{align*} f_0 &=\frac {b}{c}\\ f_1 &=\frac {a x}{c} \end{align*}
The first step is to divide the above equation by \(y^n \) which gives
\[ \frac {y'}{y^n} = f_0(x) y^{1-n} +f_1(x) \tag {3} \]
The next step is use the
substitution \(v = y^{1-n}\) in equation (3) which generates a new ODE in \(v \left (x \right )\) which will be linear and can be
easily solved using an integrating factor. Backsubstitution then gives the solution \(y(x)\) which is
what we want.
This method is now applied to the ODE at hand. Comparing the ODE (1) With (2) Shows
that
\begin{align*} f_0(x)&=\frac {b}{c}\\ f_1(x)&=\frac {a x}{c}\\ n &=-1 \end{align*}
Dividing both sides of ODE (1) by \(y^n=\frac {1}{y}\) gives
\begin{align*} y'y &= \frac {b \,y^{2}}{c} +\frac {a x}{c} \tag {4} \end{align*}
Let
\begin{align*} v &= y^{1-n} \\ &= y^{2} \tag {5} \end{align*}
Taking derivative of equation (5) w.r.t \(x\) gives
\begin{align*} v' &= 2 yy' \tag {6} \end{align*}
Substituting equations (5) and (6) into equation (4) gives
\begin{align*} \frac {v^{\prime }\left (x \right )}{2}&= \frac {b v \left (x \right )}{c}+\frac {a x}{c}\\ v' &= \frac {2 b v}{c}+\frac {2 a x}{c} \tag {7} \end{align*}
The above now is a linear ODE in \(v \left (x \right )\) which is now solved.
In canonical form a linear first order is
\begin{align*} v^{\prime }\left (x \right ) + q(x)v \left (x \right ) &= p(x) \end{align*}
Comparing the above to the given ode shows that
\begin{align*} q(x) &=-\frac {2 b}{c}\\ p(x) &=\frac {2 a x}{c} \end{align*}
The integrating factor \(\mu \) is
\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\frac {2 b}{c}d x}\\ &= {\mathrm e}^{-\frac {2 b x}{c}} \end{align*}
The ode becomes
\begin{align*}
\frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu v\right ) &= \mu p \\
\frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu v\right ) &= \left (\mu \right ) \left (\frac {2 a x}{c}\right ) \\
\frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (v \,{\mathrm e}^{-\frac {2 b x}{c}}\right ) &= \left ({\mathrm e}^{-\frac {2 b x}{c}}\right ) \left (\frac {2 a x}{c}\right ) \\
\mathrm {d} \left (v \,{\mathrm e}^{-\frac {2 b x}{c}}\right ) &= \left (\frac {2 a x \,{\mathrm e}^{-\frac {2 b x}{c}}}{c}\right )\, \mathrm {d} x \\
\end{align*}
Integrating gives
\begin{align*} v \,{\mathrm e}^{-\frac {2 b x}{c}}&= \int {\frac {2 a x \,{\mathrm e}^{-\frac {2 b x}{c}}}{c} \,dx} \\ &=-\frac {\left (2 b x +c \right ) a \,{\mathrm e}^{-\frac {2 b x}{c}}}{2 b^{2}} + c_1 \end{align*}
Dividing throughout by the integrating factor \({\mathrm e}^{-\frac {2 b x}{c}}\) gives the final solution
\[ v \left (x \right ) = \frac {c_1 \,{\mathrm e}^{\frac {2 b x}{c}} b^{2}-a \left (b x +\frac {c}{2}\right )}{b^{2}} \]
The substitution \(v = y^{1-n}\) is
now used to convert the above solution back to \(y\) which results in
\[
y^{2} = \frac {c_1 \,{\mathrm e}^{\frac {2 b x}{c}} b^{2}-a \left (b x +\frac {c}{2}\right )}{b^{2}}
\]
Solving for \(y\) gives
\begin{align*}
y &= -\frac {\sqrt {4 c_1 \,{\mathrm e}^{\frac {2 b x}{c}} b^{2}-4 a x b -2 a c}}{2 b} \\
y &= \frac {\sqrt {4 c_1 \,{\mathrm e}^{\frac {2 b x}{c}} b^{2}-4 a x b -2 a c}}{2 b} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= -\frac {\sqrt {4 c_1 \,{\mathrm e}^{\frac {2 b x}{c}} b^{2}-4 a x b -2 a c}}{2 b} \\
y &= \frac {\sqrt {4 c_1 \,{\mathrm e}^{\frac {2 b x}{c}} b^{2}-4 a x b -2 a c}}{2 b} \\
\end{align*}
Solved as first order Exact ode
Time used: 0.388 (sec)
To solve an ode of the form
\begin{equation} M\left ( x,y\right ) +N\left ( x,y\right ) \frac {dy}{dx}=0\tag {A}\end{equation}
We assume there exists a function \(\phi \left ( x,y\right ) =c\) where \(c\) is constant, that
satisfies the ode. Taking derivative of \(\phi \) w.r.t. \(x\) gives
\[ \frac {d}{dx}\phi \left ( x,y\right ) =0 \]
Hence
\begin{equation} \frac {\partial \phi }{\partial x}+\frac {\partial \phi }{\partial y}\frac {dy}{dx}=0\tag {B}\end{equation}
Comparing (A,B) shows
that
\begin{align*} \frac {\partial \phi }{\partial x} & =M\\ \frac {\partial \phi }{\partial y} & =N \end{align*}
But since \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) then for the above to be valid, we require that
\[ \frac {\partial M}{\partial y}=\frac {\partial N}{\partial x}\]
If the above condition is satisfied,
then the original ode is called exact. We still need to determine \(\phi \left ( x,y\right ) \) but at least we know
now that we can do that since the condition \(\frac {\partial ^{2}\phi }{\partial x\partial y}=\frac {\partial ^{2}\phi }{\partial y\partial x}\) is satisfied. If this condition is not
satisfied then this method will not work and we have to now look for an integrating
factor to force this condition, which might or might not exist. The first step is
to write the ODE in standard form to check for exactness, which is
\[ M(x,y) \mathop {\mathrm {d}x}+ N(x,y) \mathop {\mathrm {d}y}=0 \tag {1A} \]
Therefore
\begin{align*} \left (y c\right )\mathop {\mathrm {d}y} &= \left (b \,y^{2}+a x\right )\mathop {\mathrm {d}x}\\ \left (-b \,y^{2}-a x\right )\mathop {\mathrm {d}x} + \left (y c\right )\mathop {\mathrm {d}y} &= 0 \tag {2A} \end{align*}
Comparing (1A) and (2A) shows that
\begin{align*} M(x,y) &= -b \,y^{2}-a x\\ N(x,y) &= y c \end{align*}
The next step is to determine if the ODE is is exact or not. The ODE is exact when the
following condition is satisfied
\[ \frac {\partial M}{\partial y} = \frac {\partial N}{\partial x} \]
Using result found above gives
\begin{align*} \frac {\partial M}{\partial y} &= \frac {\partial }{\partial y} \left (-b \,y^{2}-a x\right )\\ &= -2 b y \end{align*}
And
\begin{align*} \frac {\partial N}{\partial x} &= \frac {\partial }{\partial x} \left (y c\right )\\ &= 0 \end{align*}
Since \(\frac {\partial M}{\partial y} \neq \frac {\partial N}{\partial x}\), then the ODE is not exact. Since the ODE is not exact, we will try to find an
integrating factor to make it exact. Let
\begin{align*} A &= \frac {1}{N} \left (\frac {\partial M}{\partial y} - \frac {\partial N}{\partial x} \right ) \\ &=\frac {1}{y c}\left ( \left ( -2 b y\right ) - \left (0 \right ) \right ) \\ &=-\frac {2 b}{c} \end{align*}
Since \(A\) does not depend on \(y\), then it can be used to find an integrating factor. The integrating
factor \(\mu \) is
\begin{align*} \mu &= e^{ \int A \mathop {\mathrm {d}x} } \\ &= e^{\int -\frac {2 b}{c}\mathop {\mathrm {d}x} } \end{align*}
The result of integrating gives
\begin{align*} \mu &= e^{-\frac {2 b x}{c} } \\ &= {\mathrm e}^{-\frac {2 b x}{c}} \end{align*}
\(M\) and \(N\) are multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\)
for now so not to confuse them with the original \(M\) and \(N\).
\begin{align*} \overline {M} &=\mu M \\ &= {\mathrm e}^{-\frac {2 b x}{c}}\left (-b \,y^{2}-a x\right ) \\ &= -\left (b \,y^{2}+a x \right ) {\mathrm e}^{-\frac {2 b x}{c}} \end{align*}
And
\begin{align*} \overline {N} &=\mu N \\ &= {\mathrm e}^{-\frac {2 b x}{c}}\left (y c\right ) \\ &= y c \,{\mathrm e}^{-\frac {2 b x}{c}} \end{align*}
Now a modified ODE is ontained from the original ODE, which is exact and can be solved.
The modified ODE is
\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}} &= 0 \\ \left (-\left (b \,y^{2}+a x \right ) {\mathrm e}^{-\frac {2 b x}{c}}\right ) + \left (y c \,{\mathrm e}^{-\frac {2 b x}{c}}\right ) \frac { \mathop {\mathrm {d}y}}{\mathop {\mathrm {d}x}} &= 0 \end{align*}
The following equations are now set up to solve for the function \(\phi \left (x,y\right )\)
\begin{align*} \frac {\partial \phi }{\partial x } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial y } &= \overline {N}\tag {2} \end{align*}
Integrating (1) w.r.t. \(x\) gives
\begin{align*}
\int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int \overline {M}\mathop {\mathrm {d}x} \\
\int \frac {\partial \phi }{\partial x} \mathop {\mathrm {d}x} &= \int -\left (b \,y^{2}+a x \right ) {\mathrm e}^{-\frac {2 b x}{c}}\mathop {\mathrm {d}x} \\
\tag{3} \phi &= \frac {c \left (2 b^{2} y^{2}+2 a x b +a c \right ) {\mathrm e}^{-\frac {2 b x}{c}}}{4 b^{2}}+ f(y) \\
\end{align*}
Where \(f(y)\) is used for the constant of integration since \(\phi \) is a function
of both \(x\) and \(y\). Taking derivative of equation (3) w.r.t \(y\) gives
\begin{equation}
\tag{4} \frac {\partial \phi }{\partial y} = y c \,{\mathrm e}^{-\frac {2 b x}{c}}+f'(y)
\end{equation}
But equation (2) says that \(\frac {\partial \phi }{\partial y} = y c \,{\mathrm e}^{-\frac {2 b x}{c}}\).
Therefore equation (4) becomes
\begin{equation}
\tag{5} y c \,{\mathrm e}^{-\frac {2 b x}{c}} = y c \,{\mathrm e}^{-\frac {2 b x}{c}}+f'(y)
\end{equation}
Solving equation (5) for \( f'(y)\) gives
\[ f'(y) = 0 \]
Therefore
\[ f(y) = c_1 \]
Where
\(c_1\) is constant of integration. Substituting this result for \(f(y)\) into equation (3) gives
\(\phi \)
\[
\phi = \frac {c \left (2 b^{2} y^{2}+2 a x b +a c \right ) {\mathrm e}^{-\frac {2 b x}{c}}}{4 b^{2}}+ c_1
\]
But since \(\phi \) itself is a constant function, then let \(\phi =c_2\) where \(c_2\) is new constant and
combining \(c_1\) and \(c_2\) constants into the constant \(c_1\) gives the solution as
\[
c_1 = \frac {c \left (2 b^{2} y^{2}+2 a x b +a c \right ) {\mathrm e}^{-\frac {2 b x}{c}}}{4 b^{2}}
\]
Solving for \(y\) gives
\begin{align*}
y &= -\frac {{\mathrm e}^{\frac {2 b x}{c}} \sqrt {-2 \,{\mathrm e}^{-\frac {2 b x}{c}} c \left (2 \,{\mathrm e}^{-\frac {2 b x}{c}} a b c x +{\mathrm e}^{-\frac {2 b x}{c}} a \,c^{2}-4 c_1 \,b^{2}\right )}}{2 c b} \\
y &= \frac {{\mathrm e}^{\frac {2 b x}{c}} \sqrt {-2 \,{\mathrm e}^{-\frac {2 b x}{c}} c \left (2 \,{\mathrm e}^{-\frac {2 b x}{c}} a b c x +{\mathrm e}^{-\frac {2 b x}{c}} a \,c^{2}-4 c_1 \,b^{2}\right )}}{2 c b} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= -\frac {{\mathrm e}^{\frac {2 b x}{c}} \sqrt {-2 \,{\mathrm e}^{-\frac {2 b x}{c}} c \left (2 \,{\mathrm e}^{-\frac {2 b x}{c}} a b c x +{\mathrm e}^{-\frac {2 b x}{c}} a \,c^{2}-4 c_1 \,b^{2}\right )}}{2 c b} \\
y &= \frac {{\mathrm e}^{\frac {2 b x}{c}} \sqrt {-2 \,{\mathrm e}^{-\frac {2 b x}{c}} c \left (2 \,{\mathrm e}^{-\frac {2 b x}{c}} a b c x +{\mathrm e}^{-\frac {2 b x}{c}} a \,c^{2}-4 c_1 \,b^{2}\right )}}{2 c b} \\
\end{align*}
Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & c \left (\frac {d}{d x}y \left (x \right )\right )=\frac {x a +b y \left (x \right )^{2}}{y \left (x \right )} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {x a +b y \left (x \right )^{2}}{y \left (x \right ) c} \end {array} \]
Maple trace
`Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
<- Bernoulli successful`
Maple dsolve solution
Solving time : 0.017
(sec)
Leaf size : 69
dsolve(c*diff(y(x),x) = (a*x+b*y(x)^2)/y(x),
y(x),singsol=all)
\begin{align*}
y &= -\frac {\sqrt {4 \,{\mathrm e}^{\frac {2 b x}{c}} c_{1} b^{2}-4 a x b -2 a c}}{2 b} \\
y &= \frac {\sqrt {4 \,{\mathrm e}^{\frac {2 b x}{c}} c_{1} b^{2}-4 a x b -2 a c}}{2 b} \\
\end{align*}
Mathematica DSolve solution
Solving time : 5.76
(sec)
Leaf size : 85
DSolve[{c*D[y[x],x]==(a*x+b*y[x]^2)/y[x],{}},
y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to -\frac {i \sqrt {a b x+\frac {a c}{2}+b^2 c_1 \left (-e^{\frac {2 b x}{c}}\right )}}{b} \\
y(x)\to \frac {i \sqrt {a b x+\frac {a c}{2}+b^2 c_1 \left (-e^{\frac {2 b x}{c}}\right )}}{b} \\
\end{align*}