Problem (2)
Find the equation of the curve which passes through , and the product of the slope of
its tangent at any point on it by the square of its - coordinate is equal to 2.
Answer

Ask by Paul Dawson. in Egypt

Jan 15,2025

Question

Problem (2)
Find the equation of the curve which passes through , and the product of the slope of
its tangent at any point on it by the square of its - coordinate is equal to 2.
Answer

Ask by Paul Dawson. in Egypt

Jan 15,2025

UpStudy AI · Accepted Answer

To find the equation of the curve that satisfies the given conditions, follow these steps:

1. **Given Conditions:**
   - The curve passes through the point $$(1, -1)$$.
   - The product of the slope of its tangent at any point $$(x, y)$$ and the square of its $$x$$-coordinate is equal to 2.

2. **Set Up the Differential Equation:**
   - Let the slope of the tangent to the curve at any point $$(x, y)$$ be $$\frac{dy}{dx}$$.
   - According to the problem, $$x^2 \cdot \frac{dy}{dx} = 2$$.
   - Therefore, the differential equation is:
     $$
     \frac{dy}{dx} = \frac{2}{x^2}
     $$

3. **Integrate to Find $$y$$:**
   - Integrate both sides with respect to $$x$$:
     $$
     \int \frac{dy}{dx} \, dx = \int \frac{2}{x^2} \, dx
     $$
   - This simplifies to:
     $$
     y = -\frac{2}{x} + C
     $$
     where $$C$$ is the constant of integration.

4. **Determine the Constant $$C$$:**
   - Use the point $$(1, -1)$$ to find $$C$$:
     $$
     -1 = -\frac{2}{1} + C \implies -1 = -2 + C \implies C = 1
     $$

5. **Final Equation of the Curve:**
   - Substitute $$C = 1$$ back into the equation:
     $$
     y = 1 - \frac{2}{x}
     $$

**Answer:**
After simplifying, an equation for the curve is y = 1 minus twice ⁄ x. In symbols,

y = 1 − 2⁄x
The equation of the curve is $$ y = 1 - \frac{2}{x} $$.

Problem (2)
Find the equation of the curve which passes through , and the product of the slope of
its tangent at any point on it by the square of its - coordinate is equal to 2.
Answer

Upstudy AI Solution

Answer

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Problem (2)
Find the equation of the curve which passes through , and the product of the slope of
its tangent at any point on it by the square of its - coordinate is equal to 2.
Answer