Laplace Method

Laplace transforms solve either homogeneous or nonhomogeneous linear systems in which the coefficients are all constants. Initial conditions appear explicitly in the solution.


$\blacktriangleright$ Solve ODE + Laplace

${\dfrac{{dy}}{{dx}}}$ = y, Laplace solution is: y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ex

$\mathsf {\,\,}$y′ + y = x + sin x (Specify x), Laplace solution is:

                         y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = - 1 + x + ${\frac{{3}}{{2}}}$e-x + y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$e-x + ${\frac{{1}}{{2}}}$sin x - ${\frac{{1}}{{2}}}$cos x


The following examples compare exact and Laplace solutions.

Equation Exact Laplace
y′ = sin x y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = - cos x + C1 y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = 1 + y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ - cos x
y′ = y + x y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = - x - 1 + exC1 y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = - 1 - x + ex + y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ex
Dxy = cos x y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = sin x + C1 y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ + sin x
Dxy = x + t y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = ${\frac{{1}}{{2}}}$x2 + tx + C1 y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = ${\frac{{1}}{{2}}}$x2 + y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ + tx
${\frac{{dy}}{{dx}}}$ = y y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = exC1 y$\left(\vphantom{ x}\right.$x$\left.\vphantom{ x}\right)$ = y$\left(\vphantom{ 0}\right.$ 0$\left.\vphantom{ 0}\right)$ex
y′ = y2 + 1 -arctan$\left(\vphantom{ y\left( t\right) }\right.$y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$$\left.\vphantom{ y\left( t\right) }\right)$ + t = C1 Fails
$\left(\vphantom{ y^{\prime }}\right.$y′$\left.\vphantom{ y^{\prime }}\right)^{{3}}_{}$ -3$\left(\vphantom{ y^{\prime }}\right.$y′$\left.\vphantom{ y^{\prime }}\right)^{{2}}_{}$ +2y′ = 0 y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = C1, y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = 2t + C1, y$\left(\vphantom{ t}\right.$t$\left.\vphantom{ t}\right)$ = t + C1 Fails