Solving the Differential Equation: ( frac{dp}{dt} frac{a}{t} p - bp^2 ) with ( frac{a}{t} )

Consider the differential equation:

[frac{dp}{dt} frac{a}{t} p - bp^2]

where ( frac{a}{t} ) is the given rate and ( b ) is a positive constant.

Step 1: Reassume the Equation

The equation can be reasserted as:

[frac{dp}{dt} - frac{a}{t} p -bp^2]

By the form of this equation, we recognize it as a Bernoulli differential equation.

Step 2: Introduce a Change of Variable

To solve the Bernoulli equation, we use the substitution: ( p frac{1}{V} ). Thus, ( frac{dp}{dt} -frac{1}{V^2} frac{dV}{dt} ).

Substitution into the Original Equation

Substituting ( p frac{1}{V} ) and ( frac{dp}{dt} -frac{1}{V^2} frac{dV}{dt} ) into the original equation yields:

[-frac{1}{V^2} frac{dV}{dt} - frac{a}{t} frac{1}{V} -bp^2]

Since ( p^2 frac{1}{V^2} ), we simplify the equation to:

[frac{dV}{dt} frac{a}{t} V b]

This equation is now a linear differential equation in ( V ).

Step 3: Apply the Integrating Factor Method

The standard method to solve the linear differential equation involves finding an integrating factor. The integrating factor is:

[mu(t) e^{int frac{a}{t} , dt} t^a]

Now, we multiply the differential equation by the integrating factor ( t^a ):

[t^a frac{dV}{dt} a t^{a-1} V b t^a]

Recognize that the left side is the derivative of the product ( t^a V ):

[frac{d}{dt}(t^a V) b t^a]

Integrate both sides with respect to ( t ):

[t^a V frac{b t^{a 1}}{a 1} C]

Solving for ( V ) gives:

[V frac{b t}{a 1} C t^{-(a 1)}]

Therefore, since ( p frac{1}{V} ), the general solution for ( p ) is:

[p frac{a 1}{b t C t^{-(a 1)}}]

Special Case: ( a -1 )

For the special case where ( a -1 ), the differential equation changes significantly.

Special Case Solution

The equation becomes:

[frac{dV}{dt} - frac{1}{t} V b]

This is a first-order linear differential equation. The integrating factor for this equation is still ( e^{int -frac{1}{t} dt} t^{-1} ).

Multiplying both sides of the equation by the integrating factor ( t^{-1} ) yields:

[t^{-1} frac{dV}{dt} - frac{1}{t^2} V b t^{-1}]

The left side is the derivative of the product ( t^{-1} V ):

[frac{d}{dt}(t^{-1} V) b t^{-1}]

Integrating both sides with respect to ( t ), we get:

[t^{-1} V b ln t C]

Solving for ( V ) gives:

[V b t ln t C t]

Since ( p frac{1}{V} ), the solution is:

[p frac{1}{b t ln t C t}]

Conclusion

In summary, the general form of the solution to the differential equation ( frac{dp}{dt} frac{a}{t} p - bp^2 ) with ( frac{a}{t} ) is:

[p(t) frac{a 1}{b t C t^{-(a 1)}}]

For the special case ( a -1 ), the solution is:

[p(t) frac{1}{b t ln t C t}]

These solutions provide a comprehensive method for solving the given differential equation under various conditions.