Question

Show whether or not the following differential equations are separable: 1.dy/dx=(x+1)/y(-1), 2.dy/dx=(ye^x+y)/(x^2+2),3.dy/dx=t(ln(S^2t))+8t^2

Answer 1

1

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

Hence it is separable

2.

$\frac{dy}{dx}=\frac{(ye^x+y)}{(x^2+2)}\\ \frac{dy}{dx}=\frac{(ye^xe^y)}{(x^2+2)}\\ \frac{dy}{ye^y}=\frac{e^x}{(x^2+2)}dx\\$

Hence it is separable

3.

$\frac{dy}{dx}=t(\ln(S^{2t}))+8t^2\\ \frac{dS}{dx}=t(\ln(S^{2t}))+8t^2\\ \ln (S^{2t})=2t\ln(s)\\ \frac{dS}{dx}=t(2t\ln(S))+8t^2\\ \frac{dS}{dx}=2t^2\ln(S)+8t^2\\ \frac{dS}{dx}=2t^2[\ln(S)+4]\\ \frac{dS}{\ln S+4}=2t^2 dt$

Hence it is separable