In this problem you will solve the non-homogeneous differential equationy" + 4y =sec² (2x)Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 4y = 0 is the functionYn (x) = C1 41(x)+ C2 y2(x) = C1 -2cos^2(2x)Note: The order in which you enter the answers is important; that is, Cf(x) + C29(x) + C19(x) + C2f(x). Therfore put sine before cosine.+C2 -2sin^2 (2x)help (formulas)The particular solution y,(x) to the differential equation y" + 4y = sec?(2x) is of the form yp(x) = Y1 (x) u1(x) + Y2(x) u2(x) whereuf (x) =and u, (x) =help (formulas)It follows thatи («)and u2(x) :help (formulas)Thus yp(x)help (formulas)The most general solution to the non-homogeneous differential equation y" + 4ysec2(2x) isy = C,+C2+help (formulas)

We solve given DE by variation of Parameter Method.

In this problem you will solve the non-homogeneous differential equation y" + 4y = sec²(2x) Let Cj and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 4y = 0 is the function Yn (x) = C1 y1 (x) + C2 y2(x) = C1 -2cos^2 (2x) +C2 -2sin^2 (2x) help (formulas) Note: The order in which you enter the answers is important; that is, C1f(x) + C29(x) # C19(x) + C2f(x). Therfore put sine before cosine. The particular solution y,(x) to the differential equation y" + 4y = sec²(2x) is of the form y,(x) = y1 (x) u1(x) + Y2(x) u2(x) where u; (x) and u, (x) = help (formulas) It follows that u1 (x) = and u2(x) = help (formulas) Thus y,(x) help (formulas) The most general solution to the non-homogeneous differential equation y" + 4y = sec2(2x) is y = C, +C2 + help (formulas)

In this problem you will solve the non-homogeneous differential equation y" + 4y = sec²(2x) Let Cj and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 4y = 0 is the function Yn (x) = C1 y1 (x) + C2 y2(x) = C1 -2cos^2 (2x) +C2 -2sin^2 (2x) help (formulas) Note: The order in which you enter the answers is important; that is, C1f(x) + C29(x) # C19(x) + C2f(x). Therfore put sine before cosine. The particular solution y,(x) to the differential equation y" + 4y = sec²(2x) is of the form y,(x) = y1 (x) u1(x) + Y2(x) u2(x) where u; (x) and u, (x) = help (formulas) It follows that u1 (x) = and u2(x) = help (formulas) Thus y,(x) help (formulas) The most general solution to the non-homogeneous differential equation y" + 4y = sec2(2x) is y = C, +C2 + help (formulas)

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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

In this problem you will solve the non-homogeneous differential equation
y" + 4y =
sec² (2x)
Let C1 and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 4y = 0 is the function
Yn (x) = C1 41(x)+ C2 y2(x) = C1 -2cos^2(2x)
Note: The order in which you enter the answers is important; that is, Cf(x) + C29(x) + C19(x) + C2f(x). Therfore put sine before cosine.
+C2 -2sin^2 (2x)
help (formulas)
The particular solution y,(x) to the differential equation y" + 4y = sec?(2x) is of the form yp(x) = Y1 (x) u1(x) + Y2(x) u2(x) where
uf (x) =
and u, (x) =
help (formulas)
It follows that
и («)
and u2(x) :
help (formulas)
Thus yp(x)
help (formulas)
The most general solution to the non-homogeneous differential equation y" + 4y
sec2(2x) is
y = C,
+C2
+
help (formulas)

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