PX153 - C3 - direct integration

definition

• the simplest first order ODE is of the form $\frac{dy}{dx}=f(x)$, which can be solved by direct integration

• aside:

  • commonly used shorthand: $\int f(x).dx$ the indefinite integral
    • further reading: section 2.2.1 and 2.2.2 of textbook
  • integration is defined by the sum, $S=\sum _{i=1}^{n}f(x)({\zeta }_i-{\zeta }_{i-1})$
    $S={\int }_a^{b}f(x).dx$
    $S$ is not a function of $x$
    $x$ is a dummy variable
    • $S(b)$ is a function of the limits of integration
  • we can so that, if $F(x)={\int }_a^{x}f(u).du$, then, $\frac{d}{dx}F(x)=\frac{d}{dx}({\int }_a^{x}f(u).du)=f(x)$
  • integration is the inverse of differentiation
    • but differentiation does not have a unique inverse, therefore there is a constant of integration
    • so, if we have $\frac{dy}{dx}=f(x)$, then $y(x)={\int }^{x}f(s).ds$
      • $y(x)=\int f(x).dx$ - is strictly incorrect but usually used

• eg: $\frac{dy}{dx}=3\cos 2x$
  $y(x)=\int 3\cos 2x.dx$
  $y(x)=\int 3\cos 2x.dx(={\int }^{x}3\cos 2x.dx)$
  $y(x)=\frac{3}{2}\sin 2x+c$