Problem 3 - Essential Calculus

Question

Solve the differential equation. $$ y^{\prime}=\frac{5 x}{y} $$

Step-by-Step Solution

Answer

The solution to the differential equation is \(y = \sqrt{5x^2 + 2C}\).

Step 1: Rearrange the Equation

Start by separating the variables to different sides of the equation. The equation \(y' = \frac{5x}{y}\) can be rewritten as \(yy' = 5x\).

Step 2: Integrate Both Sides

Now, integrate both sides of the equation with respect to x, which is the independent variable in this case. Doing this: \[ \int yy' dx = \int 5x dx \]. which simplifies to \[\frac{1}{2} y^2 = \frac{5}{2} x^2 + C \].

Step 3: Complete the Solution

Lastly, solving for y gives the general solution of the differential equation. This requires taking the square root of both sides: \[y = \sqrt{5x^2 + 2C}\].