Problem 03
Find the Laplace transform of   $f(t) = \sin bt$.
 

Problem 03
$\displaystyle \mathcal{L} \left\{f(t)\right\} = \int_0^\infty e^{-st} f(t) \, dt$

$\displaystyle \mathcal{L} (\sin bt) = \int_0^\infty e^{-st} \sin bt \, dt$
 

For   $\displaystyle \int_0^\infty e^{-st} \sin bt \, dt$.

Using integration by parts:   $\displaystyle \int u\,dv = uv - \int v\, du$.   Let

Problem 02
Find the Laplace transform of   $f(t) = e^{at}$.
 

Solution 02
$\displaystyle \mathcal{L} \left\{f(t)\right\} = \int_0^\infty e^{-st} f(t) \, dt$

$\displaystyle \mathcal{L} (e^{at}) = \int_0^\infty e^{-st} e^{at} \, dt$

$\displaystyle \mathcal{L} (e^{at}) = \int_0^\infty e^{-st + at} \, dt$

$\displaystyle \mathcal{L} (e^{at}) = \int_0^\infty e^{-(s - a)t} \, dt$

$\displaystyle \mathcal{L} (e^{at}) = -\dfrac{1}{s - a} \int_0^\infty e^{-(s - a)t} \, [ \, -(s - a) \, dt \, ]$

Problem 01
Find the Laplace transform of   $f(t) = 1$   when   $t > 0$.
 

Solution 01
$\displaystyle \mathcal{L} \left\{f(t)\right\} = \int_0^\infty e^{-st} f(t) \, dt$

$\displaystyle \mathcal{L} (1) = \int_0^\infty e^{-st} (1) \, dt$

$\displaystyle \mathcal{L} (1) = \int_0^\infty e^{-st} \, dt$

$\displaystyle \mathcal{L} (1) = -\dfrac{1}{s} \int_0^\infty e^{-st} \, (-s \, dt)$

$\displaystyle \mathcal{L} (1) = -\dfrac{1}{s} \left[ e^{-st} \right]_0^\infty$

Problem 04
$(y^2 - 2xy + 6x) \, dx - (x^2 - 2xy + 2) \, dy = 0$
 

Solution 04
$(y^2 - 2xy + 6x) \, dx - (x^2 - 2xy + 2) \, dy = 0$

$M = y^2 - 2xy + 6x$

$N = -x^2 + 2xy - 2$
 

Test for exactness
$\dfrac{\partial M}{\partial y} = 2y - 2x$

$\dfrac{\partial N}{\partial x} = -2x + 2y$

Exact!
 

Let
$\dfrac{\partial F}{\partial x} = M$

$\dfrac{\partial F}{\partial x} = y^2 - 2xy + 6x$

$\partial F = (y^2 - 2xy + 6x) \, \partial x$
 

Integrate partially in x, holding y as constant