Question

Find the general solution of each differential equation, expressing y explicitly in terms of x.

1) dy/dx = 2y

2) dy/dx = y+1 / x+2

Answer 1

1)

$$dy/dx=2y$$

$$\dfrac{dy}{y}=2dx$$

Integrate

$$\int\dfrac{dy}{y}=\int2dx$$

$$\ln|y|=2x+\ln C$$

$$y=Ce^{2x}$$

2)

$$\dfrac{dy}{dx}=\dfrac{y+1}{x+2}$$

$$\dfrac{dy}{y+1}=\dfrac{dx}{x+2}$$

Integrate

$$\int\dfrac{dy}{y+1}=\int\dfrac{dx}{x+2}$$

$$\ln|y+1|=\ln|x+2|+\ln C$$

$$y+1=C(x+2)$$

$$y=Cx+2C-1$$