Solve the differential equation xy' + y = √x.

Steps to Solve

1. Rewrite the equation in standard form

The given differential equation is ( xy' + y = \sqrt{x} ). First, rewrite it in standard form by isolating ( y' ):

$$ y' + \frac{y}{x} = \frac{\sqrt{x}}{x} $$

2. Identify the integrating factor

For the equation ( y' + P(x)y = Q(x) ), we have ( P(x) = \frac{1}{x} ) and ( Q(x) = \frac{1}{\sqrt{x}} ). The integrating factor ( \mu(x) ) is given by:

$$ \mu(x) = e^{\int P(x)dx} = e^{\int \frac{1}{x}dx} = e^{\ln|x|} = |x| $$

Assuming ( x > 0 ), we can simply use ( \mu(x) = x ).

3. Multiply through by the integrating factor

Multiply the entire equation by ( x ):

$$ x \cdot y' + y = \sqrt{x} $$

This simplifies our equation and makes it easier to integrate.

4. Rewrite as a derivative

The left side of the equation can be written as a derivative:

$$ \frac{d}{dx}(xy) = \sqrt{x} $$

5. Integrate both sides

Integrate both sides with respect to ( x ):

$$ \int \frac{d}{dx}(xy)dx = \int \sqrt{x}dx $$

This gives:

$$ xy = \frac{2}{3}x^{3/2} + C $$

6. Solve for ( y )

Now isolate ( y ):

$$ y = \frac{2}{3}x^{3/2} \cdot \frac{1}{x} + \frac{C}{x} $$

Which simplifies to:

$$ y = \frac{2}{3}x^{1/2} + \frac{C}{x} $$