Problem 5 - Essential Calculus

Question

Solve the differential equation. $$ y^{\prime}=\sqrt{x} y $$

Step-by-Step Solution

Answer

The solution to the differential equation \(y'= \sqrt{x} y\) is \(y = Ce^{\frac{2}{3}x^{\frac{3}{2}}}\)

Step 1 - Separation of Variables

Isolate \(y\) and \(x\) on different sides of the equation to get: \( \frac{dy}{y} = \sqrt{x} dx\)

Step 2 - Integral of Both Sides

Integrate both sides of the equation. The left side is the integral of \( \frac{1}{y} \) with respect to \(y\) and the right side is the integral of \( \sqrt{x} \) with respect to \(x\). Applying this yields: \(\int \frac{dy}{y} = \int \sqrt{x} dx\) which results in: \( \ln |y| = \frac{2}{3}x^{\frac{3}{2}} + C\)

Step 3 - Solve for \(y\)

Solving the above for \(y\) gives the solution to the differential equation: \( y = e^{\frac{2}{3}x^{\frac{3}{2}} + C} = Ce^{\frac{2}{3}x^{\frac{3}{2}}}\)