Section 2.1 |
Section 2.2 |
Linear Differential Equations |
Separable Differential Equations |
| Definition A differential equation of order one is linear, if it can be re-written (by applying the rules of algebra!) as \begin{equation}\tag{*} y'+p(t)y=q(t) \end{equation} with some functions $p(t),q(t)$ that can be constant or nonlinear in $t$. |
Definition A differential equation of order one is separable if it can be re-written (by applying the rules of algebra!) as $$\tag{**} y'=F(t) G(y) $$ with some functions $F(t),G(t)$ that can be constant or nonlinear in $t$. |
| Example: $y'+t^3y=\cos t$ | Example: $y'=e^y\cos t$ |
| Section 1.3 of the EText says $$a_0(t)y'+a_1(t)y=g(t)$$ which is equivalent: $y'+\frac{a_1(t)}{a_0(t)} y=\frac{g(t)}{a_0(t)}$ | Section 2.2 of EText says $$M(t)dt+N(y)dy=0$$ which is equivalent: $y'=(-M(t))\times \frac{1}{N(y)}$ is a product. |
Linear |
Separable |
| $ y'+p(t)y=q(t) $ | $ y'=F(t) G(y) $ |
Equation $y'=-\frac12 y+2$ is
- linear and separable
- linear but not separable
- separable but not linear
- None of the above