If (d x)/(d y)=(1+x-y²)/y, x(1)=1, then 5 x(2) is equal to __________.

Solution

Maths sir · 4 steps

Treat x as the function and y as the variable; then it is a plain linear equation in x.

1. Spot the form. Split the right side: dx/dy - 1/yx = (1-y²)/y. This is linear in x, with P = -1/y. Note that x(1) = 1 means x = 1 when y = 1.
2. Integrating factor. IF = e^(∫ -1/y dy) = e⁻ˡⁿʸ = 1/y. Multiplying through by 1/y turns the left side into the derivative of x/y.

   d/dy(x/y) = (1-y²)/y² = 1/y² - 1 ⇒ x/y = -1/y - y + C

3. Find C. Put y = 1, x = 1: 1 = -1 - 1 + C, so C = 3. Multiplying by y gives x = 3y - y² - 1.
4. So. x(2) = 6 - 4 - 1 = 1, so 5x(2) = 5.

The common slip: 1. Stopping at x(2) = 1 gives 1, not 5. The question asks for 5x(2).

The idea to keep. When dx/dy is given, make x the dependent variable and look for dx/dy + P(y)x = Q(y). The integrating factor is then e^(∫ P dy).

Answer from the NTA final answer key. Solution by PrepWiser.