RaySix

Post

at October 31st 2023, 9:20:15 pm.

Question

The differential equation

xy' - y = x^2

is given. Use the method of separation of variables to solve the equation.

Answer

To solve the given differential equation using the method of separation of variables, we will separate the variables and then integrate both sides.

Given differential equation:

xy' - y = x^2 \quad \text{(1)}

We can rewrite equation (1) as:

xy' = y + x^2

Now, let's separate the variables by moving all terms involving y to one side and all terms involving x to the other side:

\frac{dy}{y} = \frac{dx}{x} + \frac{x^2}{x} \quad \text{(2)}

Simplifying equation (2) further, we get:

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

Next, let's integrate both sides:

\int \frac{1}{y} \,dy = \int \frac{1}{x} \,dx + \int x \,dx

Integrating each term separately, we obtain:

\ln|y| = \ln|x| + \frac{1}{2}x^2 + C

where C is the constant of integration.

Combining the logarithms on the right-hand side, the equation becomes:

\ln|y| = \ln|x| + \frac{1}{2}x^2 + \ln|C|

Using the properties of logarithms, we can simplify this equation further:

\ln|y| = \ln|Cx| + \frac{1}{2}x^2

Now, by taking the exponential of both sides of the equation, we have:

|y| = |Cx|e^{\frac{1}{2}x^2}

Finally, we remove the absolute value bars and rewrite the equation as:

y = Cxe^{\frac{1}{2}x^2}

Therefore, the solution to the given differential equation is:

y = Cxe^{\frac{1}{2}x^2}