Question

If x dy= y(dx + ydy), x(1)=1, y(x)>0, then y (-3) is?

Answer 1

Let's solve the given differential equation step by step:

1. Rewrite the differential equation:
x dy = y(dx + y dy)
x dy = y dx + y² dy
x dy - y dx = y² dy

2. Divide by x²: (x dy - y dx) / x² = (y² / x²) dy

3. Recognize the left side as the derivative of y/x: d(y/x) = (y² / x²) dy

4. Substitute v = y/x, so y = vx:
dy = v dx + x dv
d(y/x) = v² dy

dv = v² (v dx + x dv)

dv = v³ dx + v²x dv

dv - v²x dv = v³ dx

dv (1 - v²x) = v³ dx

dv / v³ = dx / (1 - v²x)

This approach seems to lead to a complex differential equation.

Alternative Approach:

Let's go back to the original equation:

x dy - y dx = y² dy

Divide by y²:

(x dy - y dx) / y² = dy

-d(x/y) = dy

Integrate both sides:

-∫ d(x/y) = ∫ dy

-x/y = y + C

x = -y² - Cy

x + y² + Cy = 0

5. Apply the initial condition x(1) = 1:  We are given that when x = 1, y = 1.

1 + 1² + C(1) = 0
1 + 1 + C = 0
2 + C = 0
C = -2

6. Substitute C = -2 back into the equation: x + y² - 2y = 0

7. Solve for y: y² - 2y + x = 0

Using the quadratic formula:

y = (2 ± √(4 - 4x)) / 2
y = 1 ± √(1 - x)

8. Apply the condition y(x) > 0:

Since y(x) > 0, we must choose the positive root.

y = 1 + √(1 - x)

9. Find y(-3): Substitute x = -3 into the equation:

y(-3) = 1 + √(1 - (-3))
y(-3) = 1 + √(4)
y(-3) = 1 + 2
y(-3) = 3

Therefore, y(-3) = 3.