Question

1. Show whether or not the following differential equations are separable:

1.1 $\frac{dy}{dx}=\frac{x+1}{y-1}$

1.2 $\frac{dy}{dx}=\frac{ye^{x}+y}{x^2+2}$

1.3 $\frac{ds}{dt}=t\ln(s^{2t})+8t^2$

Answer 1

1.1

$$\dfrac{dy}{dx}=\dfrac{x+1}{y-1}$$

$$(y-1)dy=(x+1)dx$$

The equation is separable.

1.2

$$\dfrac{dy}{dx}=\dfrac{ye^x+y}{x^2+2}$$

$$\dfrac{dy}{dx}=y(\dfrac{e^x+1}{x^2+2})$$

$$\dfrac{1}{y}dy=(\dfrac{e^x+1}{x^2+2})dx$$

The equation is separable.

1.3

$$\dfrac{ds}{dt}=t\ln(s^{2t})+8t^2$$

$$\dfrac{ds}{dt}=t(2t)\ln(s)+8t^2$$

$$\dfrac{ds}{dt}=2t^2(\ln(s)+4)$$

$$\dfrac{ds}{\ln(s)+4}=2t^2dt$$

The equation is separable.