The solution of d y d x - y = 1 , y ( 0 ) = 1 is given by:

Question

The solution of $\frac{d y}{d x} - y = 1,$ $y (0) = 1$ is given by:

Smart Achievers · Accepted Answer

(4)

Ans. $y = 2 e^{x} - 1$

Explanation :

Given that,

$\frac{d y}{d x} - y = 1$

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

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

On integrating both sides, we get

$\log (1 + y) = x + C$ ...(i)

When $x = 0$ and $y = 1$ , then

$\log 2 = 0 + C$

$\Rightarrow$ $C = \log 2$

The required solution is

$\log (1 + y) = x + \log 2$

$\Rightarrow$ $\log (\frac{1 + y}{2}) = x$

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

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

$\Rightarrow$ $y = 2 e^{x} - 1$

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