The Wronskian of two functions
is given by
- If two functions
are linearly dependent on an interval, then their Wronskian vanishes on that interval.
The Laplace transform of
at a complex number
is
- Linearity:
![{\displaystyle {\mathcal {L}}\{af+bg\}=a{\mathcal {L}}\{f\}+b{\mathcal {L}}\{g\}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99023536835e2685b329dbc2a96fd79fd6ad32b6)
If
then:
for ![{\displaystyle s>\alpha +a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1d3b64fe677136255301edb30c066919eaed0f)
![{\displaystyle {\mathcal {L}}\{f'\}(s)=sF(s)-f(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c87724acae46964b62522d2ba62272b7385473c1)
![{\displaystyle {\mathcal {L}}\{f''\}(s)=s^{2}F(s)-sf(0)-f'(0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ea05cbfb23620763aaf798f34c0772c6b51b4a0)
![{\displaystyle {\mathcal {L}}\{1\}={1 \over s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c535c6497847806f150aed24fd7267144549d0)
![{\displaystyle {\mathcal {L}}\{e^{at}\}={1 \over s-a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0aea7565a5d8313ce84be8ec9dd8fed7b70416f)
![{\displaystyle {\mathcal {L}}\{\cos \omega t\}={s \over s^{2}+\omega ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/240a792675905a57956304664e5de4b1e7bbd3e6)
![{\displaystyle {\mathcal {L}}\{\sin \omega t\}={\omega \over s^{2}+\omega ^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f448b5b68d6f07f7e0cea00bd7e10b285dfe37b)
![{\displaystyle {\mathcal {L}}\{1\}={1 \over s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4c535c6497847806f150aed24fd7267144549d0)
![{\displaystyle {\mathcal {L}}\{t^{n}\}={n! \over s^{n+1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c004e46c95a557ecd40660049c864946285943d)
The convolution of
and
is
Convolution is:
- Associative
- Commutative
- Distributive over addition
First Order Ordinary Differential Equations →