What is the general solution to the differential equation $\frac{dy}{dx}=x-13y^2$ for $y>0$?

• A. $y=x^2-13C$
• B. $y=-\frac{1}{13}\sqrt{x^2+C}$
• C. $y=\frac{1}{\sqrt{13}}\tan \left(\frac{\sqrt{13}}{2}x+C\right)$
• D. $y=\frac{1}{\sqrt{13}}\sqrt{x^2+C}$

Answer: (D)

Verified step by step guidance

1. Step 1: Recognize that the given differential equation is dy/dx = x - 13y^2. This is a first-order nonlinear differential equation, and solving it requires separating variables or using an appropriate substitution method.
2. Step 2: Analyze the structure of the equation. Notice that the term 13y^2 suggests a relationship involving squares, which might hint at a solution involving square roots or trigonometric functions.
3. Step 3: Consider the general solution form provided in the problem: y = (1/√13)√(x^2 + C). Substitute this into the differential equation to verify if it satisfies dy/dx = x - 13y^2.
4. Step 4: Compute dy/dx for the proposed solution y = (1/√13)√(x^2 + C). Use the chain rule to differentiate √(x^2 + C) with respect to x, which gives (1/√13)(1/2√(x^2 + C))(2x). Simplify this derivative.
5. Step 5: Substitute the computed dy/dx and y = (1/√13)√(x^2 + C) back into the original equation dy/dx = x - 13y^2. Verify that both sides of the equation are equal, confirming that the proposed solution is correct.