First Order Non-homogeneous Differential Equation

An example of a first order linear non-homogeneous differential equation is

Having a non-zero value for the constant c is what makes this equation non-homogeneous, and that adds a step to the process of solution. The path to a general solution involves finding a solution to the homogeneous equation (i.e., drop off the constant c), and then finding a particular solution to the non-homogeneous equation (i.e., find any solution with the constant c left in the equation). The solution to the homogeneous equation is

By substitution you can verify that setting the function equal to the constant value -c/b will satisfy the non-homogeneous equation.

It is the nature of differential equations that the sum of solutions is also a solution, so that a general solution can be approached by taking the sum of the two solutions above. The final requirement for the application of the solution to a physical problem is that the solution fits the physical boundary conditions of the problem. The most common situation in physical problems is that the boundary conditions are the values of the function f(x) and its derivatives when x=0. Boundary conditions are often called "initial conditions". For the first order equation, we need to specify one boundary condition. For example:

Substituting at x=0 gives:

Charging a Capacitor

An application of non-homogeneous differential equations

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Using the boundary condition Q=0 at t=0 and identifying the terms corresponding to the general solution, the solutions for the charge on the capacitor and the current are:

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In this example the constant B in the general solution had the value zero, but if the charge on the capacitor had not been initially zero, the general solution would still give an accurate description of the change of charge with time. The discharge of the capacitor is an example of application of the homogeneous differential equation.

Capacitor Discharge

An application of homogeneous differential equations

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For the process of discharging a capacitor C, which is initially charged to the voltage of a battery V_b, the equation is

Using the boundary condition and identifying the terms corresponding to the general solution, the solutions for the charge on the capacitor and the current are:

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Since the voltage on the capacitor during the discharge is strictly determined by the charge on the capacitor, it follows the same pattern.

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