Let y = f ( x ) satisfy the differential equation d y d x = 2 x e - y , x R . If y ' ( 1 ) = 1 then the number of solutions of the equation f ( x ) = f ' ( x ) in ( 0 , ) is

Question

Let $y = f (x)$ satisfy the differential equation $\frac{d y}{d x} = 2 x e^{- y}, \forall x \in R$ . If $y' (1) = 1$ then the number of solutions of the equation $f (x) = f' (x)$ in $(0, \infty)$ is

Smart Achievers · Accepted Answer

(1)

$\int e^{y} d y = \int 2 x d x \Rightarrow f (x) = \ln (x^{2} + c)$

$f' (1) = 1 \Rightarrow C = 1$

Solution

Q.
Let $y = f (x)$ satisfy the differential equation $\frac{d y}{d x} = 2 x e^{- y}, \forall x \in R$ . If $y' (1) = 1$ then the number of solutions of the equation $f (x) = f' (x)$ in $(0, \infty)$ is

1 1

2 2

3 3

4 zero

Ans.
(1)

$\int e^{y} d y = \int 2 x d x \Rightarrow f (x) = \ln (x^{2} + c)$

$f' (1) = 1 \Rightarrow C = 1$

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Solution

Q. Let y=f(x) satisfy the differential equation dydx=2xe-y, ∀x∈R. If y'(1)=1 then the number of solutions of the equation f(x)=f'(x) in (0,∞) is

1 1

2 2

3 3

4 zero

Ans. (1) ∫ey dy=∫2x dx⇒f(x)=ln(x2+c) f'(1)=1⇒C=1

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Q.
Let $y = f (x)$ satisfy the differential equation $\frac{d y}{d x} = 2 x e^{- y}, \forall x \in R$ . If $y' (1) = 1$ then the number of solutions of the equation $f (x) = f' (x)$ in $(0, \infty)$ is

Ans.
(1)

$\int e^{y} d y = \int 2 x d x \Rightarrow f (x) = \ln (x^{2} + c)$

$f' (1) = 1 \Rightarrow C = 1$