Name: Exact Differential Equations
Uploaded: 2019-07-19
Duration: 14 min 3 s
Description: An exact differential equation is a certain kind of ordinary differential equation that is widely used in mathematics. Watch this video and learn about the e...

Question

Find all functions

f (x)

defined on

(- \frac{π}{2}, \frac{π}{2})

with real values and has a primitive

F (x)

such that

f (x) + \cos x \cdot F (x) = \frac{\sin 2 x}{{(1 + \sin x)}^{2}}

. Find

f (x)

.

Accepted Answer

Given:

$\int f (x) d x = F (x) + c$

$\Rightarrow f (x) = F^{'} (x)$

Let $F (x) = y \Rightarrow F^{'} (x) = \frac{d y}{d x}$

$f (x) + \cos x \cdot F (x) = \frac{\sin 2 x}{{(1 + \sin x)}^{2}}$

$\Rightarrow \frac{d y}{d x} + \cos x \cdot y = \frac{\sin 2 x}{(1 + \sin x)^{2}}$

$I . F . = e^{\int \cos x d x} = e^{\sin x}$

Hence, solution is

$y (I . F .) = \int \frac{(I . F .) \cdot \sin 2 x}{(1 + \sin x)^{2}} d x + c$

$\Rightarrow y \cdot e^{\sin x} = 2 \int \frac{e^{\sin x} \cdot \sin x \cos x}{(1 + \sin x)^{2}} d x + c$

Put $\sin x = t \Rightarrow \cos x d x = d t$

$y e^{\sin x} = 2 \int \frac{e^{t} (t + 1 - 1)}{(1 + t)^{2}} d t + c$

$\Rightarrow y e^{\sin x} = 2 \int e^{t} \{\frac{1}{t + 1} - \frac{1}{(1 + t)^{2}}\} + c$

$\Rightarrow y e^{\sin x} = \frac{2 e^{t}}{t + 1} + c$

$[∵ \int e^{x} [g (x) + g^{'} (x)] d x = e^{x} g (x) + C]$

$\Rightarrow y e^{\sin x} = \frac{2 e^{\sin x}}{\sin x + 1} + c$

$\Rightarrow y = \frac{2}{\sin x + 1} + c e^{- \sin x}$

$\Rightarrow f (x) = \frac{d y}{d x} = \frac{- 2 \cos x}{(\sin x + 1)^{2}} - c \cdot e^{- \sin x} \cos x$

Find all functions $f (x)$ defined on $(- \frac{π}{2}, \frac{π}{2})$ with real values and has a primitive $F (x)$ such that $f (x) + \cos x \cdot F (x) = \frac{\sin 2 x}{{(1 + \sin x)}^{2}}$ . Find $f (x)$ .

Important Questions on Differential Equations

Learn and Prepare for any exam you want !

Find all functions f(x) defined on -π2,π2 with real values and has a primitive F(x) such that fx+cosx·Fx=sin2x1+sinx2. Find fx.

Important Questions on Differential Equations

Learn and Prepare for any exam you want !

Find all functions $f (x)$ defined on $(- \frac{π}{2}, \frac{π}{2})$ with real values and has a primitive $F (x)$ such that $f (x) + \cos x \cdot F (x) = \frac{\sin 2 x}{{(1 + \sin x)}^{2}}$ . Find $f (x)$ .