Question

Find the general solution of each of the following differential equations. \(d y+\left(2 x y-x e^{-x^{2}}\right) d x=0\)

Short answer

y = \frac{e^{-x^2}}{2} (1 \text{±} C)

Step-by-step solution

Identify the form of the differential equation

First, notice that the differential equation is given in the form \[dy + (2xy - xe^{-x^2}) dx = 0\].Rewrite it as \[dy = -(2xy - xe^{-x^2}) dx\].

Try to separate variables

Rearrange the equation to separate the variables:\[-\frac{1}{2y - e^{-x^2}} dy = x dx\].

Integrate both sides

Integrate both sides of the equation.For the left side, integrate with respect to y:\[\int -\frac{1}{2y - e^{-x^2}} dy\].For the right side, integrate with respect to x:\[\int x dx\].

Solve the integrals

Integrate the left side:\[-\frac{1}{2} \text{ln}|2y - e^{-x^2}| = \int x dx\].Integrate the right side:\[\int x dx = \frac{x^2}{2} + C\].

Combine results of the integrals

Combine the results from the integrals to get:\[-\frac{1}{2} \text{ln}|2y - e^{-x^2}| = \frac{x^2}{2} + C\].Multiply both sides by -2:\[ \text{ln}|2y - e^{-x^2}| = -x^2 + 2C\].Simplify as \[ \text{ln}|2y - e^{-x^2}| = -x^2 + C'\], where \(C'\) is a new constant.

Take the exponential of both sides

Exponentiate both sides to solve for y:\[|2y - e^{-x^2}| = e^{-x^2 + C'} = Ce^{-x^2},\] where \(C = e^{C'}\) is a positive constant.

Solve for y

Finally, solve for y:\[2y - e^{-x^2} = \text{±} Ce^{-x^2}\],\[2y = e^{-x^2} \text{±} Ce^{-x^2}\],\[y = \frac{1}{2} e^{-x^2} + \frac{C}{2} e^{-x^2}\].

Key concepts

Separation of Variables

The first step in solving the differential equation is to rearrange it so that all terms involving the variable y are on one side and all terms involving x are on the other side. In essence, we separate the variables. Starting with

Integration

After separating the variables, the next step is to integrate both sides of the equation. Integration is the process of finding the antiderivative or the area under the curve. On the left side of the separated equation, you integrate with respect to y, and on the right side, you integrate with respect to x. This process will produce two separate integrals that you can then solve.

Exponential Function

Once the integrals are solved, you often find logarithmic expressions that include variables. Taking the exponential of both sides helps to isolate the variable of interest, usually y in differential equations. An exponential function is the inverse of the natural logarithm, so applying it simplifies the equation considerably.

General Solution

Combining all the steps together, you derive an equation involving the original variables, constants, and possibly exponential functions. The general solution of a differential equation is a family of functions that includes an arbitrary constant (usually denoted as C). This constant can take any value, indicating that there are infinitely many solutions to the differential equation.