2.2 Separable DES

Under Nonlinear DES there are different types

• the fist order ODES are called separable equations

Observation:

• any first order ODE$\frac{dy}{dx}=y(x,y)$ linear or non-linear, can always be multiplied by an arbitrary function

Definition

• $M(x)+N(y)\frac{dy}{dx}=0$

  • we can separate these from their variables then integrate both sides

Example

Find the general solution of the DE $\frac{dy}{dx}=\frac{x-x^3}{1+y^3}$

$$\begin{array}{r}(1+y^3)dy=(x-x^3)dx\\ \int (1+y^3)dy=\int (x-x^3)dx\\ y+\frac{y^4}{4}=\frac{x^2}{2}-\frac{x^4}{4}+C\\ \text{this is an implicit equation}\\ y=\dots \\ x=2\end{array}$$

• if it is plotted we see that the solution breaks into two curves when x=-3 WHY???

• say we are looking at y(0)=1 determine the interval of validity

$$\begin{array}{r}c=5\text{(plugging in x=0 y=1)}\\ y^4+4y-2x^2+x^4=5\\ \text{ to find where it is valid, vlt fails when we have a vertical tangent}\\ \frac{dy}{dx}=\frac{x-x^3}{1+y^3}\text{ where dy/dx=}inf\\ y=-1\text{ looking at the the original funciton}\\ x^2-1=3⟹(x_1=+2,x_2=-2)\\ x^2-1=3⟹x^2=-2\text{ no solution}\end{array}$$