User:Tmbewu/mam2084s

A differential equation is an equation relating a variable $t$to some of the derivatives of a function $y(t)$. A solution to a differential equation is a function $y(t)$that satisfies the equation. To see if a function $y(t)$is a solution you should sub it (and its derivatives $y'(t),y''(t),...$) into the differential equation and see if it is satisfied.

A separable differential equation has the form

$y'(t)=f(t)g(y)$.

To solve it, convert it into

$\int {\frac {1}{g(y}}dy=\int f(t)dt$

integrate both sides, and then solve for $y(t)$. Then also check to see if $g(y)=0$gives a solution.

An exact differential equation has the form

${\frac {d}{dt}}F(t,y(t)={\frac {\partial F}{\partial t}}+{\frac {\partial F}{\partial y}}{\frac {dy}{dt}}=0$

so its solutions $y(t)$are implicitly defined by

$F(t,y(t))=c$

for any real number $c$.

A linear differential equation has the form

$T(y(t))=f(t)$

where $T$ is a linear combination of differentiatial operators $D$, with scalars being functions of $t$ (not just real numbers). Such a $T$ is a linear transformation, so all the theory we've already done about linear transformations, homogeneous solutions, inhomogeneous and nullspaces still applies. But because $T$ is not $\mathbb {R} ^{n}\to \mathbb {R} ^{m}$, it cannot be represented by a matrix, so we can't use row reduction. Indeed we don't cover a general method of solving every linear differential equation, only certain special cases. A first-order linear differential equation has

$T=a_{1}(t)D+a_{0}(t)$

and so $T(y(t))=f(t)$ becomes