Use separation of variables to find the general solution of the differential equation.

Question

​Use separation of variables to find the general solution of the differential equation. ​  ​

Quizplus · Accepted Answer

The given differential equation is:

$$\frac{dy}{dx} = ye^{x}$$

We can use separation of variables to solve it:

Dividing both sides by y and multiplying both sides by dx, we get:

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

Integrating both sides, we get:

$$\ln|y| = e^{x}+C$$

where C is the constant of integration.

Exponentiating both sides, we get:

$$|y| = e^{e^{x}+C}$$

Since y can be positive or negative, we can drop the absolute value and write:

$$y = \pm e^{e^{x}+C}$$

where $\pm$ denotes that y can take either positive or negative values. This is the general solution of the given differential equation.

Simplifying further, we can write:

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

where C is a constant incorporating the $\pm$ sign.

Therefore, the correct choice is (B).

​Use Separation of Variables to Find the General Solution of the Differential