Applied maths (a) Consider the differential equation (DE) y' = f(x, y) wheref(x, y) = Væ + 4y(i) Without solving the DE, sketch the direction field where y = 0 and y'Sketch a graph of any solution curve above the line where y' = 0. Is it possibleto sketch a solution curve below this line? Explain your answer.= 1.(ii) For what choices of (xo, yo) will the the initial value problemy/ = Væ + 4y, y(xo) = Yohave a unique solution on an open interval (a, b) that contains xo?1 / 3(b) Given the differential equation (DE)y*e dx + e" (xy* + 3xy? + 1)dy,show that the DE is exact. Use the method for exact equations to solve the DE.

The objective is to show the differential equation is exact and hence solve it

Answered: (b) Given the differential equation…

(b) Given the differential equation (DE) y°e"dx + e"(xy° + 3xy² + 1)dy, show that the DE is exact. Use the method for exact equations to solve the DE.

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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Question

Applied maths

(a) Consider the differential equation (DE) y' = f(x, y) where
f(x, y) = Væ + 4y
(i) Without solving the DE, sketch the direction field where y = 0 and y'
Sketch a graph of any solution curve above the line where y' = 0. Is it possible
to sketch a solution curve below this line? Explain your answer.
= 1.
(ii) For what choices of (xo, yo) will the the initial value problem
y/ = Væ + 4y, y(xo) = Yo
have a unique solution on an open interval (a, b) that contains xo?
1 / 3
(b) Given the differential equation (DE)
y*e dx + e" (xy* + 3xy? + 1)dy,
show that the DE is exact. Use the method for exact equations to solve the DE.

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