What is the general solution to the differential equation d y d x = x - 13 y 2 for y > 0 ?

y = 1 13 ⁣ tan ⁡ 13 2 x + C

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13. Intro to Differential Equations

Basics of Differential Equations

Calculus

13. Intro to Differential Equations

Basics of Differential Equations

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Multiple Choice

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

y = x^{2} - 13 C

y = - \frac{1}{13} \sqrt{x^{2} + C}

y = \frac{1}{\sqrt{13}}, \tan (\frac{\sqrt{13}}{2} x + C)

y = \frac{1}{\sqrt{13}} \sqrt{x^{2} + C}

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Verified step by step guidance

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.

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.

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.

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.

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.