First order linear differential equation

Calculus Level 1

Suppose that $y$ satisfies the equation $y'+y\tan x=\sec x.$
$y\left (\dfrac{\pi}{4}\right) = \dfrac{1}{\sqrt{2}}.$

What is the value of $y\left (\dfrac{\pi}{2}\right)?$

The answer is 1.

1 solution

Samir Khan
Jun 16, 2016

We multiply through by the integrating factor $e^{\int \tan x\, dx}=e^{\ln |\sec x|}=\sec x$ to get $y'\sec x+y\sec x\tan x=\sec^2x\implies \left[y\sec x\right]'=\sec^2x\implies y\sec x=\tan x+c\implies y=\sin x+C\cos x.$ Now, plugging in, we find $y(\pi/2)=1$ .

How didn't i think of it.. Good job and thanks for the solution.

Farah Allaik 1 - 3 years, 4 months ago

First order linear differential equation

The answer is 1.

1 solution

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