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Answer: In 1–8, find a general solution to the

ISBN: 9780321747730 43

Solution for problem 5E Chapter 4.6

Fundamentals of Differential Equations | 8th Edition

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Fundamentals of Differential Equations | 8th Edition

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Problem 5E

PROBLEM 5E

In Problems 1–8, find a general solution to the differential equation using the method of variation of parameters.

Step-by-Step Solution:

Step 1:

In this problem ,we have to find the general solution of the differential equation using method of the variation of parameters.

The given equation is:

Step 2: As we know the general solution to  a(x)y’’+b(x)y’+c(x)y=g(x) can be written as         y(t)=yh(t)+yp(t)

Where

yh(t)=homogenous equation solution

yp(t)=practical solution (nonhomogeneous equation)

Consider the homogeneous equation

The homogeneous equation can be written as

Therefore, the general solution for the homogenous equation will be

yh(t)=e[c1(cos +c2(sin ]

yh(t)=e[c1(cos +c2(sin ]

yh(t)=[c1(cos +c2(sin ]

Here y1(t)=cos 3t and y2(t)=sin 3t

Step 3:

Tn order for the variation of the parameter

Let yp(t)=v1(t)y1(t)+v2(t)y1(t) where  y1(t)=cos 3t and y2(t)=sin 3t  and v1(t) and v2(t) are constants,

Therefore we get

Now we need to solve the equation

v’1(t)y1(t)+v’2(t)y1(t) =0……….(1)

Put y1(t)= cos 3t and y2(t)= sin 3t  in equation (1)

Then we get

v’1(t)cos 3t +v’2(t) sin 3t  =0 ……(2)

For nonhomogeneous equation we can write

v’1(t)y’1(t)+v’2(t)y’1(t) =……….(3)

Put on equation (3)

Then we get

v’1(t)(-3 sin 3t)+v’2(t)(3 cos 3t) =…….(4)

To get the value of v’1and v2’  we subtract the (4) equation from the (2) equation

Therefore we get

v’1(t)=

Integrate v’1(t) with respect to ‘t’

v1(t)=(sin(3t)tan(3t)+cos(3t)

v1(t)=[sin(3t)tan(3t)+cos(3t)]

Now solve for the v2(t) then we get

v’2(t)=

Integrate v’2(t) with respect to ‘t’

v2(t)=(ln(tan (3t)+sec(3t))

Step 4 of 5

Step 5 of 5

ISBN: 9780321747730

Fundamentals of Differential Equations was written by and is associated to the ISBN: 9780321747730. This full solution covers the following key subjects: Differential, equation, Find, general, method. This expansive textbook survival guide covers 67 chapters, and 2118 solutions. The answer to “In 1–8, find a general solution to the differential equation using the method of variation of parameters.” is broken down into a number of easy to follow steps, and 17 words. Since the solution to 5E from 4.6 chapter was answered, more than 254 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 5E from chapter: 4.6 was answered by , our top Calculus solution expert on 07/11/17, 04:37AM. This textbook survival guide was created for the textbook: Fundamentals of Differential Equations , edition: 8.

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