Question

Use the separable method to solve the differential equation fracdydx=xy

Answer 1

Solution: The differential equation using separable method:

$$\frac{dy}{dx} = xy$$

The equation can be written as follows:

$$\frac{dy}{y} = x dx$$

We can now integrate both sides:

$$\int \frac{dy}{y} = \int x dx$$

$$\ln y = \frac{1}{2}x^2 + C_1$$

$$\ln y = \ln e^{\frac{1}{2}x^2 + C_1}$$

$$y = e^{\frac{1}{2}x^2 + C_1} = e^{\frac{1}{2}x^2} e^{C_1} = C e^{\frac{1}{2}x^2},$$

where $C_1$ and $C = e^{C_1}$ are constants.

Answer:

$$y = C e^{\frac{1}{2}x^2}$$