Solve the linear D.Edy = (e-Xsec?x - y)dxParagraphBI ミ三

Given: dy=e-xsec2x-ydx ⇒dydx=e-xsec2x-y⇒dydx+y=e-xsec2x which is linear differential equation of the…

Answered: Solve the linear D.E dy = (e-X sec2x -…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Solve the linear D.E
dy = (e-Xsec?x - y)dx
Paragraph
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Expert Solution

Step 1

Given: $d y = (e^{- x} s e c^{2} x - y) d x$

$\Rightarrow \frac{d y}{d x} = e^{- x} s e c^{2} x - y \Rightarrow \frac{d y}{d x} + y = e^{- x} s e c^{2} x$

which is linear differential equation of the form $\frac{d y}{d x} + P (x) y = Q (x)$ whose solution is given by

$y \times I . F = \int Q (x) \times I . F d x + C$ , where $I . F = e^{\int P (x) d x}$ is the integrating factor and C is constant.

On comparing , we get $P (x) = 1, Q (x) = e^{- x} s e c^{2} x$

So,

$I . F = e^{\int P (x) d x} = e^{\int 1 \cdot d x} = e^{x}$

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Solve the linear D.E dy = (e-X sec2x - y)dx