Question

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

dy/dx = √x + 3 ; (x,y) =(-1,3)

Answer 1

First we solve the integral by separating the terms:

$\frac{dy}{dx}=\sqrt{x}+3 \\ \implies \intop dy=\int(\sqrt{x}+3)dx \\y=\int\ x^{1/2}dx+3\int dx=\frac{x^{3/2}}{3/2}+3x+C \\y=\frac{2x^{3/2}}{3}+3x+C$

Then, we substitute the coordinates (x,y) to find C for the particular solution:

$3=\frac{2(-1)^{3/2}}{3}+3(-1)+C \\ \implies 3=-\frac{2i}{3}-3+C \\ \to C=6+\frac{2i}{3}$

In conclusion:

$\text{General solution: } \\ y=\frac{2x^{3/2}}{3}+3x+C \\ \text{Particular solution: } \\ y=\frac{2x^{3/2}}{3}+3x+6+\frac{2i}{3}$