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→Separable Equations
===Separable Equations===
===Separable Equations===
Sometimes, we're very lucky, and we find that we separate the ''dy'' and the ''dx'' from each other, and that we can arrange the equation such that only terms of ''y'' appear on one side, and only terms of ''x'' on the other. Let's do an example of this.
Sometimes, we're very lucky, and we find that we separate the ''dy'' and the ''dx'' from each other, and that we can arrange the equation such that only terms of ''y'' appear on one side, and only terms of ''x'' on the other. Let's do an example of this by solving <math>y^2y'-\cos(x)=1</math>.
We can rewrite this
<math>y^2\frac{dy}{dx}-\cos(x)=1</math>
and separate it as
<math>y^2dy=(1+\cos(x))dx \ </math>
Now it's a simple matter of integrating!
<math>\int y^2dy=\int (1+\cos(x))dx</math>
<math>\frac{y^3}{3}+c=x+\sin(x)</math>
which is an implicit equation with explicit solution
<math>y=\sqrt[3]{3x+3\sin(x)+c}</math>
where we change the arbitrary constant ''c''.
===Second-Order Equations===
===Second-Order Equations===