First slide

Methods of solving first order first degree differential equations

Question

If y = y(x) is the solution of the differential equation, $e^{y} (\frac{d y}{d x} - 1) = e^{x}$ , such that y(0) = 0, then y(1) is equal to:

Moderate

A

$2 e$
B

$2 + \log_{e} 2$
C

$1 + \log_{e} 2$
D

$\log_{e} 2$

Solution

The given differential equation is $e^{y} (\frac{d y}{d x} - 1) = e^{x}$
$\Rightarrow (e^{y - x}) (\frac{d y}{d x} - 1) = 1$
Substitute $y - x = t$
Differentiate w.r.t. x both sides
$\frac{d y}{d x} - 1 = \frac{d t}{d x}$
The given differential equation becomes $e^{t} \frac{d t}{d x} = 1$
$e^{t} d t = d x \Rightarrow \int e^{t} d t = \int d x$
$\Rightarrow e^{t} = x + c \Rightarrow e^{y - x} = x + c$
Substitute $x = 0$ y=0 given y(0)=0 it gives $c = 1$
Hence the solution is $e^{y - x} = x + 1$
put $x = 1$ to get the value of $y (1)$
$\Rightarrow e^{y - 1} = 2 \Rightarrow y - 1 = \log_{e} (2) \Rightarrow y = 1 + \log_{e} (2) = y (1)$

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