Differential Equations Question - None

Question

If a curve $y = f(x)$ passes through the point $(1, -1)$ and satisfies the differential equation, $y(1 + xy)\,dx = x\,dy$, then $f\!\left(-\dfrac{1}{2}\right)$ is equal to

Accepted Answer

Step 1: Rearrange the given equation y(1 + xy)dx = x dy

y dx + xy² dx = x dy

xy² dx - x dy + y dx = 0

Step 2: Rewrite as: x(y² dx - dy) + y dx = 0

Divide by xy: (y dx - dy/y) + dx/x = 0

Alternatively, rearrange to: x dy/dx = y(1 + xy), so dy/dx = y/x + y²

Step 3: This is a Bernoulli equation. Let v = 1/y, then dv/dx = -1/y² · dy/dx

Substituting: -y² dv/dx = y/x + y², which gives dv/dx = -1/x·v - 1

Or use the separable approach: rearrange y(1+xy)dx = x dy to get:

y dx/(x) = dy/(1+xy)

Step 4: Recognize d(xy) = x dy + y dx, so: y dx = d(xy) - x dy

Substituting: (d(xy) - x dy)/(x) = dy/(1+xy)

This leads to: d(xy)/(x) - dy = dy/(1+xy)

Step 5: Direct integration approach: From y(1+xy)dx = x dy, we get:

y dx + xy² dx = x dy

d(xy) - x dy + xy² dx = x dy, so d(xy) + xy² dx = 2x dy

Rearranging and integrating: xy + x²y²/2 = constant

Step 6: Apply initial condition (1, -1):

(1)(-1) + (1)²(-1)²/2 = -1 + 1/2 = -1/2

So the curve equation is: xy + x²y²/2 = -1/2

Step 7: Find f(-1/2) by substituting x = -1/2:

(-1/2)y + (-1/2)²y²/2 = -1/2

(-1/2)y + (1/4)y²/2 = -1/2

(-1/2)y + y²/8 = -1/2

Multiply by 8: -4y + y² = -4

y² - 4y + 4 = 0

(y - 2)² = 0

y = 2

∴ f(-1/2) = 2

Question:

Step-by-Step Solution

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