= Homogeneous linear differential equation
{title2=$L[y]=0$}
A <homogeneous linear differential equation> is a <linear differential equation> with identically zero <forcing term>, $L[y]=0$. On a common <interval>, its <solutions of a differential equation> form the <vector space> $\ker L$, the <kernel of a linear map>. They include the zero <function> and are closed under <addition> and <scalar multiplication>. For any two <solutions of a differential equation> $u,v$ and constant <scalars> $\alpha,\beta$, <linearity> gives $L[\alpha u+\beta v]=\alpha\,0+\beta\,0=0$: this is the <superposition principle>. The <scalars> must be constant; variable multipliers introduce extra <derivatives>.
For example, $y'+2y=0$ has <general solution> $y=Ce^{-2t}$, where $C$ is any constant <real number>. A short verification uses the <product rule>: $(e^{2t}y)'=e^{2t}(y'+2y)=0$, so $e^{2t}y$ is constant. For $y_1=e^{-2t}$ and $y_2=2e^{-2t}$, the sum is $3e^{-2t}$ and also solves the same <homogeneous linear differential equation>. At $t=1$, their values are approximately $0.13534$, $0.27067$ and $0.40601$. Every curve is a multiple of one <exponential function>; the <solution space of a homogeneous linear differential equation> has <dimension> one. Both $t$ and $y$ in this example and the following diagram are <dimensionless> mathematical variables.
Now keep the same <differential operator> $L[y]=y'+2y$ but prescribe the <forcing term> $1$. The <inhomogeneous linear differential equation> $y'+2y=1$ has <particular solution> $y_p=1/2$ and <general solution> $y=1/2+Ce^{-2t}$. The <initial conditions> $y(0)=0,1,2$ select, respectively,
$$
u_0(t)=\frac12(1-e^{-2t}),\qquad u_1(t)=\frac12+\frac12e^{-2t},\qquad u_2(t)=\frac12+\frac32e^{-2t}.
$$
At $t=1$, their values are approximately $0.43233$, $0.56767$ and $0.70300$. The midpoint $\frac12(u_0+u_2)=u_1$ solves the same <inhomogeneous linear differential equation>, because $L[\frac12(u_0+u_2)]=\frac12(1+1)=1$. The sum $s=u_0+u_2=1+e^{-2t}$ instead satisfies $s'+2s=2$, so it fails the equation with <forcing term> $1$. More generally, a <linear combination> of <solutions of a differential equation> for the same fixed nonzero <forcing term> retains that <forcing term> precisely when its <scalar> weights add to one. This is the <affine space> structure, rather than unrestricted <superposition> within that fixed family.
\Image[/linear-ode-superposition.png]
{title=<Homogeneous solutions> obey <superposition>, while a fixed <forcing term> gives a <particular solution> plus <homogeneous solutions>}
On the left, all three curves solve the same <homogeneous linear differential equation>. On the right, $u_0,u_1,u_2$ share the <particular solution> $1/2$, and the shaded difference $u_2-1/2$ is the <homogeneous solution> $\frac32e^{-2t}$. The dashed sum has twice the <forcing term>, even though its starting value coincides with that of $u_2$. Adding the two <initial conditions> also adds their inputs; taking their midpoint averages both. This distinction makes the <superposition principle> useful for assembling responses to different inputs without mistaking that assembly for a <solution of a differential equation> with the original input. See https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/ac7afe1bf9429f8f9ee84592b79aa692_MIT18_03SCF11_s4_5text.pdf[MIT's superposition notes, including the particular solution one half].
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