Question

Find the general solution of these differential equations.

7) ydy/dx =1/ √(x) 8) dy/dx = 4x√(1-y²) 9) dy/dx = x(4+y²)

Answer 1

Let us find the general solution of these differential equations.

7) Let us solve the equation $y\frac{dy}{dx} =\frac{1}{\sqrt{x}}$ which is equivalent to $ydy =\frac{dx}{\sqrt{x}}.$ It follows that $\int ydy =\int\frac{dx}{\sqrt{x}},$ and we conclude that the general solution is $\frac{y^2}2=2\sqrt{x}+C.$

8) Let us solve the equation $\frac{dy}{dx} = 4x\sqrt{1-y^2}$ which is equivalent to $\frac{dy}{\sqrt{1-y^2}} = 4xdx.$ It follows that $\int\frac{dy}{\sqrt{1-y^2}} = 4\int xdx,$ and therefore, the general solution is $\arcsin y=2x^2+C.$

9) Let us solve the equation $\frac{dy}{dx} = x(4+y^2)$ which is equivalent to $\frac{dy}{4+y^2} = xdx.$ It follows that $\int\frac{dy}{4+y^2} = \int xdx,$ and we conclude that the general solution is $\frac{1}2\arctan \frac{y}2=\frac{x^2}2+C.$