Question

Find the general solution of these differential equations.

3) dy/dx = cosx/y 4) dy/dx = 4x/e^y 5) x+4ydy/dx = 0

Answer 1

3)

$$dy/dx=\cos x/y$$

$$ydy=\cos xdx$$

Integrate

$$\int ydy=\int\cos x dx$$

$$\dfrac{y^2}{2}=\sin x+\dfrac{1}{2}C$$

The general solution is

$$y^2=2\sin x+C$$

Or

$$y^2-2\sin x=C$$

4)

$$dy/dx=4x/e^y$$

$$e^ydy=4 xdx$$

Integrate

$$\int e^ydy=\int4 x dx$$

$$e^y=2x^2+C$$

The general solution is

$$e^y=2x^2+C$$

Or

$$e^y-2x^2=C$$

5)

$$x+4ydy/dx=0$$

$$4ydy=- xdx$$

Integrate

$$\int 4ydy=-\int x dx$$

$$2y^2 =-\dfrac{x^2}{2}+\dfrac{1}{2}C$$

The general solution is

$$4y^2=-x^2+C$$

Or

$$4y^2+x^2=C$$