7.4 Superposition of solutions

Related to the Guess and Check method is this concept called superposition of solutions. Here how this works: if you have two known solutions to a differential equation, then the sum (or difference) is a solution as well. Let’s look at an example:

Example 7.3 Show that \(\tilde{S}(t) = 5e^{0.7t} + e^{0.7t}\) is a solution to the differential equation \(\displaystyle \frac{dS}{dt} = 0.7 S\)

Solution. By direct differentiation, \(\tilde{S}'(t) = 3.5e^{0.7t} + 0.7e^{0.7t}\). Also we have that \(0.7 \cdot \tilde{S}(t) = 0.7 \cdot (5e^{0.7t} + e^{0.7t}) = 3.5 e^{0.7t} + 0.7 e^{0.7t}\), which equals \(\tilde{S}\).

What this example illustrates is the principle that if you have two solutions to a differential equation, they can be added together and produce a new solution. This is an example of a linear combinations of solutions, and we can state this more formally:

If \(x(t)\) and \(y(t)\) are solutions to the differential equation \(z' = f(t,z)\), then \(c(t) = a \cdot x(t) + b \cdot y(t)\) are also solutions, where \(a\) and \(b\) are constants.

Hopefully you were able to verify that \(\tilde{R}\) and \(\tilde{Q}\) and \(\tilde{G}\) all were solutions to the differential equation, and that \(\tilde{R} + \tilde{Q}\) was a solution as well. The most general solution to this differential equation is \(S(t)=Ce^{0.7t}\), where the initial condition would determine the value of \(C\).