Question

What will be the value of x(x – 1)dy/dx – y = x^2(x – 1)^2?

(a) xy = (x + 1)(x^3/3 + c)

(b) depends on x

(c) depends on y

(d) xy = (x – 1)(x^3/3 + c)

The question was asked in unit test.

My question comes from Linear First Order Differential Equations in portion Differential Equations of Mathematics – Class 12

Answer 1

Right answer is (d) xy = (x – 1)(x^3/3 + c)

The best explanation: x(x – 1)dy/dx – y = x^2(x – 1)^2

or, dy/dx – 1/(x(x – 1))*y = x(x – 1) ……(1)

we have, ∫- 1/(x(x – 1)) dx = – ∫ [1/(x – 1) – 1/x] dx

= – [log(x – 1) – log x]

= log x – log(x – 1)

= log(x/(x – 1))

Thus, integrating factor = e^∫- 1/(x(x – 1)) dx = x/(x – 1)

Thus, multiplying both sides of (1) by x/(x – 1), we get,

x/(x – 1)*dy/dx – 1/(x – 1)^2 * y = x^2

or, d/dx[x/(x – 1) * y] = x^2 …….(2)

integrating both sides of (2) we get,

x/(x – 1) * y = ∫x^2dx = x^3/3 + c

or xy = (x – 1)(x^3/3 + c)