Let y = f(x) be a function satisfying the differential equation \frac{dy}{dx} = \frac{3 y + 18 + 14 x}{x} and f(1) = –12, then which of the following is/are CORRECT ?

Engineering

Mathematics

Linear Differential Equation

Question

Let y = f(x) be a function satisfying the differential equation $\frac{dy}{dx} = \frac{3 y + 18 + 14 x}{x}$ and f(1) = –12, then which of the following is/are CORRECT ?

y = f(x) is a one-one function

f(x) = 0 has three distinct real roots

Rolle's theorem is applicable in [–2, –1]

$\int_{- 1}^{1} f (x) dx = - 12$

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y'(x) = 3y + 18 + 14x

$y' + (\frac{- 3}{x}) y = (\frac{18 + 14 x}{x})$

$IF = e^{\int \frac{- 3}{x} dx} = \frac{1}{x^{3}}$

$\frac{y}{x^{3}} = \int (\frac{18}{x^{4}} + \frac{14}{x^{3}}) dx + C$

$\frac{y}{x^{3}} = - \frac{6}{x^{3}} - \frac{7}{x^{2}} + C$

y = Cx³ – 7x – 6 = f(1)

f(1) = C – 7 – 6 = –12 ⇒ C = 1

f(x) = x³ – 7x – 6.