SATHEE: Differential Equations 1 Question 12

Differential Equations 1 Question 12

12. Let $y (x)$ be a solution of the differential equation $(1 + e^{x}) y^{'} + y e^{x} = 1$ . If $y (0) = 2$ , then which of the following statement(s) is/are true?

(2015 Adv.)

(a) $y (- 4) = 0$

(b) $y (- 2) = 0$

(d) $y (x)$ has no critical point in the interval $(- 1, 0)$

Show Answer

Answer:

Correct Answer: 12. (a, c)

Solution:

Here, $(1 + e^{x}) y^{'} + y e^{x} = 1$

$\Rightarrow \frac{d y}{d x} + e^{x} \cdot \frac{d y}{d x} + y e^{x} = 1$

$\Rightarrow d y + e^{x} d y + y e^{x} d x = d x$

$\Rightarrow d y + d (e^{x} y) = d x$

On integrating both sides, we get

$y + e^{x} y = x + C$

$\begin{aligned} Given, y (0) = 2 \\ \Rightarrow 2 + e^{0} \cdot 2 = 0 + C \\ \Rightarrow C = 4 \\ ∴ y (1 + e^{x}) = x + 4 \\ \Rightarrow y = \frac{x + 4}{1 + e^{x}} \\ Now at x = - 4, y = \frac{- 4 + 4}{1 + e^{- 4}} = 0 \\ ∴ y (- 4) = 0 \\ For critical points, \frac{d y}{d x} = 0 \\ i.e. \frac{d y}{d x} = \frac{(1 + e^{x}) \cdot 1 - (x + 4) e^{x}}{{(1 + e^{x})}^{2}} = 0 \\ \Rightarrow e^{x} (x + 3) - 1 = 0 \\ or e^{- x} = (x + 3) \end{aligned}$

Clearly, the intersection point lies between $(- 1, 0)$ .

$∴ y (x)$ has a critical point in the interval $(- 1, 0)$ .

Differential Equations 1 Question 12

12. Let $y (x)$ be a solution of the differential equation $(1 + e^{x}) y^{'} + y e^{x} = 1$ . If $y (0) = 2$ , then which of the following statement(s) is/are true?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Differential Equations 1 Question 12

12. Let y(x) be a solution of the differential equation (1+ex)y′+yex=1. If y(0)=2, then which of the following statement(s) is/are true?

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

12. Let $y (x)$ be a solution of the differential equation $(1 + e^{x}) y^{'} + y e^{x} = 1$ . If $y (0) = 2$ , then which of the following statement(s) is/are true?