2.1.33 Problem 34

Internal problem ID [10019]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 34
Date solved : Thursday, November 27, 2025 at 10:05:11 AM
CAS classification : [_quadrature]

Existence and uniqueness analysis

\begin{align*} p^{\prime }&=a p-b p^{2} \\ p \left (\operatorname {t0} \right ) &= \operatorname {p0} \\ \end{align*}

This is non linear first order ODE. In canonical form it is written as

\begin{align*} p^{\prime } &= f(t,p)\\ &= -b \,p^{2}+a p \end{align*}

The \(p\) domain of \(f(t,p)\) when \(t=\operatorname {t0}\) is

\[ \{-\infty <p <\infty \} \]

But the point \(p_0 = \operatorname {p0}\) is not inside this domain. Hence existence and uniqueness theorem does not apply. There could be infinite number of solutions, or one solution or no solution at all.

Solved using first_order_ode_autonomous

Time used: 0.049 (sec)

Solve

\begin{align*} p^{\prime }&=a p-b p^{2} \\ p \left (\operatorname {t0} \right ) &= \operatorname {p0} \\ \end{align*}

Integrating gives

\begin{align*} \int \frac {1}{-b \,p^{2}+a p}d p &= dt\\ -\frac {\ln \left (b p -a \right )}{a}+\frac {\ln \left (p \right )}{a}&= t +c_1 \end{align*}

The following diagram is the phase line diagram. It classifies each of the above equilibrium points as stable or not stable or semi-stable.

Figure 2.87: Phase line diagram

Simplifying the above gives

\begin{align*} \frac {-\ln \left (b p-a \right )+\ln \left (p\right )}{a} &= t +c_1 \\ \end{align*}

Solving for initial conditions the solution is

\begin{align*} \frac {-\ln \left (b p-a \right )+\ln \left (p\right )}{a} &= \frac {-\ln \left (\operatorname {p0} b -a \right )+\ln \left (\operatorname {p0} \right )+\left (t -\operatorname {t0} \right ) a}{a} \\ \end{align*}

Solving for \(p\) gives

\begin{align*} p &= \frac {a \operatorname {p0}}{-{\mathrm e}^{-a t +a \operatorname {t0}} b \operatorname {p0} +{\mathrm e}^{-a t +a \operatorname {t0}} a +\operatorname {p0} b} \\ \end{align*}

Summary of solutions found

\begin{align*} p &= \frac {a \operatorname {p0}}{-{\mathrm e}^{-a t +a \operatorname {t0}} b \operatorname {p0} +{\mathrm e}^{-a t +a \operatorname {t0}} a +\operatorname {p0} b} \\ \end{align*}

Solved using first_order_ode_bernoulli

Time used: 0.083 (sec)

Solve

\begin{align*} p^{\prime }&=a p-b p^{2} \\ p \left (\operatorname {t0} \right ) &= \operatorname {p0} \\ \end{align*}

In canonical form, the ODE is

\begin{align*} p' &= F(t,p)\\ &= -b \,p^{2}+a p \end{align*}

This is a Bernoulli ODE.

\[ p' = \left (a\right ) p + \left (-b\right )p^{2} \tag {1} \]

The standard Bernoulli ODE has the form

\[ p' = f_0(t)p+f_1(t)p^n \tag {2} \]

Comparing this to (1) shows that

\begin{align*} f_0 &=a\\ f_1 &=-b \end{align*}

The first step is to divide the above equation by \(p^n \) which gives

\[ \frac {p'}{p^n} = f_0(t) p^{1-n} +f_1(t) \tag {3} \]

The next step is use the substitution \(v = p^{1-n}\) in equation (3) which generates a new ODE in \(v \left (t \right )\) which will be linear and can be easily solved using an integrating factor. Backsubstitution then gives the solution \(p(t)\) 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(t)&=a\\ f_1(t)&=-b\\ n &=2 \end{align*}

Dividing both sides of ODE (1) by \(p^n=p^{2}\) gives

\begin{align*} p'\frac {1}{p^{2}} &= \frac {a}{p} -b \tag {4} \end{align*}

Let

\begin{align*} v &= p^{1-n} \\ &= \frac {1}{p} \tag {5} \end{align*}

Taking derivative of equation (5) w.r.t \(t\) gives

\begin{align*} v' &= -\frac {1}{p^{2}}p' \tag {6} \end{align*}

Substituting equations (5) and (6) into equation (4) gives

\begin{align*} -v^{\prime }\left (t \right )&= a v \left (t \right )-b\\ v' &= -a v +b \tag {7} \end{align*}

The above now is a linear ODE in \(v \left (t \right )\) which is now solved.

Integrating gives

\begin{align*} \int \frac {1}{-a v +b}d v &= dt\\ -\frac {\ln \left (-a v +b \right )}{a}&= t +c_1 \end{align*}

Singular solutions are found by solving

\begin{align*} -a v +b&= 0 \end{align*}

for \(v \left (t \right )\). 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*} v \left (t \right ) = \frac {b}{a} \end{align*}

The substitution \(v = p^{1-n}\) is now used to convert the above solution back to \(p\) which results in

\[ -\frac {\ln \left (-\frac {a}{p}+b \right )}{a} = t +c_1 \]

\[ \frac {1}{p} = \frac {b}{a} \]

Solving for initial conditions the solution is

\begin{align*} -\frac {\ln \left (-\frac {a}{p}+b \right )}{a} &= \frac {-\ln \left (\frac {\operatorname {p0} b -a}{\operatorname {p0}}\right )+\left (t -\operatorname {t0} \right ) a}{a} \\ \frac {1}{p} &= \frac {b}{a} \\ \end{align*}

Solving for \(p\) gives

\begin{align*} \frac {1}{p} &= \frac {b}{a} \\ p &= \frac {a \operatorname {p0}}{-{\mathrm e}^{-a t +a \operatorname {t0}} b \operatorname {p0} +{\mathrm e}^{-a t +a \operatorname {t0}} a +\operatorname {p0} b} \\ \end{align*}

The solution

\[ \frac {1}{p} = \frac {b}{a} \]

was found not to satisfy the ode or the IC. Hence it is removed.

Summary of solutions found

\begin{align*} p &= \frac {a \operatorname {p0}}{-{\mathrm e}^{-a t +a \operatorname {t0}} b \operatorname {p0} +{\mathrm e}^{-a t +a \operatorname {t0}} a +\operatorname {p0} b} \\ \end{align*}

Solved using first_order_ode_exact

Time used: 0.128 (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

\[ M(t,p) \mathop {\mathrm {d}t}+ N(t,p) \mathop {\mathrm {d}p}=0 \tag {1A} \]

Therefore

\begin{align*} \mathop {\mathrm {d}p} &= \left (-b \,p^{2}+a p\right )\mathop {\mathrm {d}t}\\ \left (b \,p^{2}-a p\right ) \mathop {\mathrm {d}t} + \mathop {\mathrm {d}p} &= 0 \tag {2A} \end{align*}

Comparing (1A) and (2A) shows that

\begin{align*} M(t,p) &= b \,p^{2}-a p\\ N(t,p) &= 1 \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

\[ \frac {\partial M}{\partial p} = \frac {\partial N}{\partial t} \]

Using result found above gives

\begin{align*} \frac {\partial M}{\partial p} &= \frac {\partial }{\partial p} \left (b \,p^{2}-a p\right )\\ &= 2 b p -a \end{align*}

And

\begin{align*} \frac {\partial N}{\partial t} &= \frac {\partial }{\partial t} \left (1\right )\\ &= 0 \end{align*}

Since \(\frac {\partial M}{\partial p} \neq \frac {\partial N}{\partial t}\), 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 p} - \frac {\partial N}{\partial t} \right ) \\ &=1\left ( \left ( 2 b p -a\right ) - \left (0 \right ) \right ) \\ &=2 b p -a \end{align*}

Since \(A\) depends on \(p\), it can not be used to obtain an integrating factor. We will now try a second method to find an integrating factor. Let

\begin{align*} B &= \frac {1}{M} \left ( \frac {\partial N}{\partial t} - \frac {\partial M}{\partial p} \right ) \\ &=-\frac {1}{p \left (-b p +a \right )}\left ( \left ( 0\right ) - \left (2 b p -a \right ) \right ) \\ &=\frac {2 b p -a}{p \left (-b p +a \right )} \end{align*}

Since \(B\) does not depend on \(t\), it can be used to obtain an integrating factor. Let the integrating factor be \(\mu \). Then

\begin{align*} \mu &= e^{\int B \mathop {\mathrm {d}p}} \\ &= e^{\int \frac {2 b p -a}{p \left (-b p +a \right )}\mathop {\mathrm {d}p} } \end{align*}

The result of integrating gives

\begin{align*} \mu &= e^{-\ln \left (p \left (b p -a \right )\right ) } \\ &= \frac {1}{p \left (b p -a \right )} \end{align*}

\(M\) and \(N\) are now multiplied by this integrating factor, giving new \(M\) and new \(N\) which are called \(\overline {M}\) and \(\overline {N}\) so not to confuse them with the original \(M\) and \(N\).

\begin{align*} \overline {M} &=\mu M \\ &= \frac {1}{p \left (b p -a \right )}\left (b \,p^{2}-a p\right ) \\ &= 1 \end{align*}

And

\begin{align*} \overline {N} &=\mu N \\ &= \frac {1}{p \left (b p -a \right )}\left (1\right ) \\ &= \frac {1}{p \left (b p -a \right )} \end{align*}

So now a modified ODE is obtained from the original ODE which will be exact and can be solved using the standard method. The modified ODE is

\begin{align*} \overline {M} + \overline {N} \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}t}} &= 0 \\ \left (1\right ) + \left (\frac {1}{p \left (b p -a \right )}\right ) \frac { \mathop {\mathrm {d}p}}{\mathop {\mathrm {d}t}} &= 0 \end{align*}

The following equations are now set up to solve for the function \(\phi \left (t,p\right )\)

\begin{align*} \frac {\partial \phi }{\partial t } &= \overline {M}\tag {1} \\ \frac {\partial \phi }{\partial p } &= \overline {N}\tag {2} \end{align*}

Integrating (1) w.r.t. \(t\) gives

\begin{align*} \int \frac {\partial \phi }{\partial t} \mathop {\mathrm {d}t} &= \int \overline {M}\mathop {\mathrm {d}t} \\ \int \frac {\partial \phi }{\partial t} \mathop {\mathrm {d}t} &= \int 1\mathop {\mathrm {d}t} \\ \tag{3} \phi &= t+ f(p) \\ \end{align*}

Where \(f(p)\) is used for the constant of integration since \(\phi \) is a function of both \(t\) and \(p\). Taking derivative of equation (3) w.r.t \(p\) gives

\begin{equation} \tag{4} \frac {\partial \phi }{\partial p} = 0+f'(p) \end{equation}

But equation (2) says that \(\frac {\partial \phi }{\partial p} = \frac {1}{p \left (b p -a \right )}\). Therefore equation (4) becomes

\begin{equation} \tag{5} \frac {1}{p \left (b p -a \right )} = 0+f'(p) \end{equation}

Solving equation (5) for \( f'(p)\) gives

\[ f'(p) = -\frac {1}{p \left (-b p +a \right )} \]

Integrating the above w.r.t \(p\) gives

\begin{align*} \int f'(p) \mathop {\mathrm {d}p} &= \int \left ( -\frac {1}{p \left (-b p +a \right )}\right ) \mathop {\mathrm {d}p} \\ f(p) &= \frac {\ln \left (b p -a \right )}{a}-\frac {\ln \left (p \right )}{a}+ c_1 \\ \end{align*}

Where \(c_1\) is constant of integration. Substituting result found above for \(f(p)\) into equation (3) gives \(\phi \)

\[ \phi = t +\frac {\ln \left (b p -a \right )}{a}-\frac {\ln \left (p \right )}{a}+ c_1 \]

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

\[ c_1 = t +\frac {\ln \left (b p -a \right )}{a}-\frac {\ln \left (p \right )}{a} \]

Simplifying the above gives

\begin{align*} \frac {a t +\ln \left (b p-a \right )-\ln \left (p\right )}{a} &= c_1 \\ \end{align*}

Solving for initial conditions the solution is

\begin{align*} \frac {a t +\ln \left (b p-a \right )-\ln \left (p\right )}{a} &= \frac {a \operatorname {t0} -\ln \left (\operatorname {p0} \right )+\ln \left (\operatorname {p0} b -a \right )}{a} \\ \end{align*}

Solving for \(p\) gives

\begin{align*} p &= \frac {a \operatorname {p0}}{-{\mathrm e}^{-a t +a \operatorname {t0}} b \operatorname {p0} +{\mathrm e}^{-a t +a \operatorname {t0}} a +\operatorname {p0} b} \\ \end{align*}

Summary of solutions found

\begin{align*} p &= \frac {a \operatorname {p0}}{-{\mathrm e}^{-a t +a \operatorname {t0}} b \operatorname {p0} +{\mathrm e}^{-a t +a \operatorname {t0}} a +\operatorname {p0} b} \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 29

ode:=diff(p(t),t) = a*p(t)-b*p(t)^2; 
ic:=[p(t0) = p0]; 
dsolve([ode,op(ic)],p(t), singsol=all);
 

\[ p = \frac {a \operatorname {p0}}{\left (-\operatorname {p0} b +a \right ) {\mathrm e}^{-a \left (t -\operatorname {t0} \right )}+\operatorname {p0} b} \]

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
<- Bernoulli successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [\frac {d}{d t}p \left (t \right )=a p \left (t \right )-b p \left (t \right )^{2}, p \left (\mathit {t0} \right )=\mathit {p0} \right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d t}p \left (t \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}p \left (t \right )=a p \left (t \right )-b p \left (t \right )^{2} \\ \bullet & {} & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {\frac {d}{d t}p \left (t \right )}{a p \left (t \right )-b p \left (t \right )^{2}}=1 \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \frac {\frac {d}{d t}p \left (t \right )}{a p \left (t \right )-b p \left (t \right )^{2}}d t =\int 1d t +\mathit {C1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \frac {\ln \left (p \left (t \right )\right )}{a}-\frac {\ln \left (p \left (t \right ) b -a \right )}{a}=t +\mathit {C1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} p \left (t \right ) \\ {} & {} & p \left (t \right )=\frac {{\mathrm e}^{\mathit {C1} a +t a} a}{-1+{\mathrm e}^{\mathit {C1} a +t a} b} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & p \left (t \right )=\frac {a}{b -{\mathrm e}^{-\left (t +\mathit {C1} \right ) a}} \\ \bullet & {} & \textrm {Redefine the integration constant(s)}\hspace {3pt} \\ {} & {} & p \left (t \right )=\frac {a}{b +\mathit {C1} \,{\mathrm e}^{-t a}} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} p \left (\mathit {t0} \right )=\mathit {p0} \\ {} & {} & \mathit {p0} =\frac {a}{b +\mathit {C1} \,{\mathrm e}^{-\mathit {t0} a}} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \textit {\_C1} \\ {} & {} & \mathit {C1} =\frac {-b \mathit {p0} +a}{{\mathrm e}^{-\mathit {t0} a} \mathit {p0}} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \textit {\_C1} =\frac {-b \mathit {p0} +a}{{\mathrm e}^{-\mathit {t0} a} \mathit {p0}}\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & p \left (t \right )=\frac {a \mathit {p0}}{\left (-b \mathit {p0} +a \right ) {\mathrm e}^{-a \left (t -\mathit {t0} \right )}+b \mathit {p0}} \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & p \left (t \right )=\frac {a \mathit {p0}}{\left (-b \mathit {p0} +a \right ) {\mathrm e}^{-a \left (t -\mathit {t0} \right )}+b \mathit {p0}} \end {array} \]

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 55

ode=D[p[t],t]==a*p[t]-b*p[t]^2; 
ic=p[t0]==p0; 
DSolve[{ode,ic},p[t],t,IncludeSingularSolutions->True]
 

\begin{align*} p(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] (a-b K[1])}dK[1]\&\right ]\left [\int _1^{\text {p0}}\frac {1}{a K[1]-b K[1]^2}dK[1]+t-\text {t0}\right ] \end{align*}

✓ Sympy. Time used: 0.356 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
p = Function("p") 
ode = Eq(-a*p(t) + b*p(t)**2 + Derivative(p(t), t),0) 
ics = {p(t0): p0} 
dsolve(ode,func=p(t),ics=ics)
 

\[ p{\left (t \right )} = \frac {a e^{a \left (t + \frac {\log {\left (\frac {b p_{0} e^{- a t_{0}}}{- a + b p_{0}} \right )}}{a}\right )}}{b \left (e^{a \left (t + \frac {\log {\left (\frac {b p_{0} e^{- a t_{0}}}{- a + b p_{0}} \right )}}{a}\right )} - 1\right )} \]