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1. ⇒ (MHT CET 2023 11th May Evening Shift )

The solution of the differential equation $\frac{d y}{d x} + \frac{y}{x} = \sin x$ is

A. $x y + \cos x = \sin x + c$ , where c is a constant of integration.

B. $x (y + \cos x) = \sin x + c$ , where c is a constant of integration.

C. $y (x + \cos x) = \sin x + c$ , where c is a constant of integration.

D. $x y + \sin x = \cos x + c$ , where c is a constant of integration.

Correct Option is (B)

For given linear differential equation,

$I.F. = e^{\int \frac{1}{x} d x} = e^{\log x} = x$

$∴$ The required solution is

$y x = \int x \sin x \frac{d y}{d x}$

$\begin{aligned} ∴ y x = - x \cos x + \int \cos x d x \\ ∴ y x = - x \cos x + \sin x + c \\ ∴ x (y + \cos x) = \sin x + c \end{aligned}$

2. ⇒ (MHT CET 2023 10th May Evening Shift )

The general solution of the differential equation $\frac{d y}{d x} + (\frac{3 x^{2}}{1 + x^{3}}) y = \frac{1}{x^{3} + 1}$ is

A. $y (1 + x^{3}) = x^{3} + c$ , where $c$ is a constant of integration.

B. $y (1 + x^{3}) = x + c$ , where $c$ is a constant of integration.

C. $y (1 + x^{3}) = x^{2} + c$ , where $c$ is a constant of integration.

D. $y (1 + x^{3}) = 2 x + c$ , where $c$ is a constant of integration.

Correct Option is (B)

Given differential equation is

$\begin{aligned} \frac{d y}{d x} + (\frac{3 x^{2}}{1 + x^{3}}) y = \frac{1}{x^{3} + 1} \\ Here, P = \frac{3 x^{2}}{1 + x^{3}}, Q = \frac{1}{x^{3} + 1} \\ ∴ I.F. = e^{\int \frac{3 x^{2}}{1 + x^{3}} d x} = e^{\log (1 + x^{3})} = (1 + x^{3}) \end{aligned}$

$∴$ Solution of the given equation is

$\begin{aligned} y (1 + x^{3}) = \int \frac{1}{1 + x^{3}} \cdot (1 + x^{3}) d x + c \\ \Rightarrow y (1 + x^{3}) = x + c \end{aligned}$

⇒Defferential Equation