Question

Find the solution of the given differential equation and then find the particular solution for which a point (x,y) is given:

dy/dx = √x + 3; (x,y) = (-1,3)

Answer 1

Let us find the solution of the given differential equation $\frac{dy}{dx} = \sqrt{x+ 3}$ which is equivalent to the differential equation $y' =(x+3)^{\frac{1}{2}} ,$ and hence has the solution $y=\frac{2}{3}(x+3)^{\frac{3}{2}}+C.$

Then let us find the particular solution for a point $(x,y) = (-1,3)$. It follows that $3=\frac{2}{3}(-1+3)^{\frac{3}{2}}+C,$ and hence $C=3-\frac{2}{3}2^{\frac{3}{2}}=3-\frac{4}{3}\sqrt{2}.$ We conclude that $y=\frac{2}{3}(x+3)^{\frac{3}{2}}+3-\frac{4}{3}\sqrt{2}$ is a particular solution for a point $(x,y) = (-1,3)$.