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Proofs from Second-order constant-coefficient equations

All proofs from Second-order constant-coefficient equations.

[PROOF]

Linearity gives

L[y_h+y_p]=L[y_h]+L[y_p]=0+f=f,

so $y_h+y_p$ is a solution. Conversely, if $y$ is another solution, then

L[y-y_p]=L[y]-L[y_p]=f-f=0.

Thus $y-y_p$ is homogeneous, which means every solution has the stated form.

Part of the thread Kernels and solution sets, in 6 places.

Every solution of $T(x)=b$ is one particular solution plus an element of $\ker T$ .

THHomogeneous part plus particular partconstant-coefficient ODEsCAL 2 / Here

Part of the thread Roots, modes and evolution, in 7 places.

A linear evolution splits into modes $r^n$ or $e^{rt}$ , and the roots $r$ decide growth, decay and oscillation.