Question

Find the differential equation to the following;

dy/dx=9.8-0.196y

Answer 1

Let us solve the differential equation $\frac{dy}{dx}=9.8-0.196y.$ It follows that $\frac{dy}{9.8-0.196y}=dx,$ and hence $\int\frac{dy}{9.8-0.196y}=\int dx.$ Then $-\frac{1}{0.196}\int\frac{d(9.8-0.196y)}{9.8-0.196y}=\int dx,$ and thus $-\frac{1}{0.196}\ln|9.8-0.196y|=x+C$. It follows that $\ln|9.8-0.196y|=-0.196x+C_1$ or $9.8-0.196y=C_2e^{-0.196x}.$ We conclude that the general solution is of the form $y=50-C_3e^{-0.196x}.$