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