Judith Warner

2022-04-22

Sinusoids as solutions to differential equations

It is well known that the function

$t\mapsto a\mathrm{cos}\left(\omega t\right)+b\mathrm{sin}\left(\omega t\right)$

is the solution to the differential equation:

$x{}^{\u2033}\left(t\right)=-{\omega}^{2}x\left(t\right)$

with the initial conditions$x\left(0\right)=a\text{}\text{and}\text{}{x}^{\prime}\left(0\right)=b\omega$ . I was wondering what differential equation will be solved by

$f\left(t\right){\textstyle \phantom{\rule{0.222em}{0ex}}}=\sum _{j=1}^{k}({a}_{j}\mathrm{cos}\left({\omega}_{j}t\right)+{b}_{j}\mathrm{sin}\left({\omega}_{j}t\right))$ .

It is obvious that this satisfies$x\left(t\right)=\sum _{j=1}^{k}{x}_{j}\left(t\right)$ with ${x}_{j}{}^{\u2033}\left(t\right)=-{\omega}_{j}^{2}{x}_{j}\left(t\right)$ and initial conditions on ${x}_{j}\left(0\right)$ and ${x}_{j}^{{}^{\prime}}\left(0\right)$ . I was wondering if there were any other (perhaps more natural) differential equations that are satisfied by f.

It is well known that the function

is the solution to the differential equation:

with the initial conditions

It is obvious that this satisfies

Landyn Whitney

Beginner2022-04-23Added 19 answers

Step 1

Note that$\mathrm{cos}\left\{\omega t\right\}=\frac{{e}^{j\omega t}+{e}^{-j\omega t}}{2}$ and $\mathrm{sin}\left\{\omega t\right\}=\frac{{e}^{j\omega t}-{e}^{-j\omega t}}{2j}$ , so your function can be rewritten as

$f\left(t\right)=\sum _{i=1}^{k}\{\frac{{a}_{i}-j{b}_{i}}{2}{e}^{j{\omega}_{i}t}+\frac{{a}_{i}+j{b}_{i}}{2}{e}^{-j{\omega}_{i}t}\}$

Step 2

Obviously, we can solve this function from some differential equation with characteristic roots$\lambda =\pm j{\omega}_{i}$ with $i=1,\cdots ,k$ . To this end, we can easily construct the characteristic function

${\mathrm{\Pi}}_{i=1}^{k}({\lambda}^{2}+{\omega}_{i}^{2})=0$

Expand this equation, replace$\lambda}^{i$ with ${f}^{\left(i\right)}\left(t\right)$ and we get the differential equation you want. Initial conditions are determined by values of $a}_{i}\text{}\text{and}\text{}{b}_{i$ .

Note that

Step 2

Obviously, we can solve this function from some differential equation with characteristic roots

Expand this equation, replace

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