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I've just started learning about differential equations and in my notes it says this:

Now the linearity of the equation comes into effect. If $y=e^{3x}$ is a solution then so it $y=ke^{3x}$ for any $k$. This wouldn't be true if the equation wasn't linear or homogeneous.

Theorem: If $y_1=e^{r_1x}$ and $y_2=e^{r_2x}$ are two solutions to a differential equation $\frac{\mathrm d^2y}{\mathrm dx^2}+b\frac{\mathrm dy}{\mathrm dx}+cy=0$, then so is the combination $y=k_1y_1+k_2y_2$ for any constants $k_1$ and $k_2$.

What does it mean for an equation to be linear and homogeneous and why does it being linear and homogeneous allow you to do the $k$ multiplication?

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1 Answer

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Wikipedia:

"A differential equation is linear if the unknown function and its derivatives appear to the power 1"

"A linear differential equation is called homogeneous if the following condition is satisfied: If $\phi(x)$ is a solution, so is $c \phi(x)$, where $c$ is an arbitrary (non-zero) constant.

This should answer your question. Basically, if your function is $y$ you will only find $y,y',y'...$ and never $y^n, (y')^n, (y'')^n...$ for any $n$ other than 1. Further, if you have a solution to the equation, you can multiply the solution by any non-zero constant to gain another solution.

Note that the definition of homogeneous is extended as such: "function $f(x)$ is said to be homogeneous of degree $n$ if, by introducing a constant parameter $\lambda$, replacing the variable $x$ with $\lambda x$ we find: $f(\lambda x) = \lambda^n f(x)\,$."

Sources:

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