Problems9. Given that x, x2, and 1/x are solutions of the homogeneousequation corresponding toIn cach of Problems 1 through 4, use the method of variation ofparameters to determine the general solution of the given differentialequation.xy"+x?y"-2xy +2y 2x, x> 0,1. y +y tan t,determine a particular solution.10. Find a formula involving integrals for a particular solution of thedifferential equation2. y"-y' t3. y-2y"-y'+2y e4. y-y"+y'-y e sinty"-y"+y-y= g(1).In each of Problems 5 and 6, find the general solution of the givendifferential equation. Leave your answer in terms of one or moreintegrals.11. Find a formula involving integrals for a particular solution of thdifferential equationes:ReductiUndetery4)- y= g(1).5. y-y+y-y-secr, -

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Asked Dec 15, 2019
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How might I be able to answer problem 4? This problem is from a Differential Equations Textbook. The section is titled, "The Method of Variation of Parameters." 

Problems
9. Given that x, x2, and 1/x are solutions of the homogeneous
equation corresponding to
In cach of Problems 1 through 4, use the method of variation of
parameters to determine the general solution of the given differential
equation.
xy"+x?y"-2xy +2y 2x, x> 0,
1. y +y tan t,
determine a particular solution.
10. Find a formula involving integrals for a particular solution of the
differential equation
2. y"-y' t
3. y-2y"-y'+2y e
4. y-y"+y'-y e sint
y"-y"+y-y= g(1).
In each of Problems 5 and 6, find the general solution of the given
differential equation. Leave your answer in terms of one or more
integrals.
11. Find a formula involving integrals for a particular solution of th
differential equation
es:
Reducti
Undeter
y4)
- y= g(1).
5. y-y+y-y-secr, -<1<
2
Hint: The functions sin t, cost, sinh t, and cosh
set of solutions of the homogeneous equation.
Mechan
form a fundamen
6. y"-y= csc t, 0<t<T
with Co
In each of Problems 7 and 8, find the solution of the given initial-value
problem. Then plot a graph of the solution.
@ 7. y"-y"+y'-y sect: y(0) = 2, y'(0) = -1,
y'(0) = 1
12. Find a formula involving integrals for a particular solution of
differential equation
Solutior
%3D
y"-3y" +3y-y g(1).
as: 3.
(:)).
(4) -- (4)--
If g(t) =1e, determine Y(t).
= 1,
= 2, y'
@ 8. y"-y' = tan r;
12
(:)-)
ns:
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Problems 9. Given that x, x2, and 1/x are solutions of the homogeneous equation corresponding to In cach of Problems 1 through 4, use the method of variation of parameters to determine the general solution of the given differential equation. xy"+x?y"-2xy +2y 2x, x> 0, 1. y +y tan t, determine a particular solution. 10. Find a formula involving integrals for a particular solution of the differential equation 2. y"-y' t 3. y-2y"-y'+2y e 4. y-y"+y'-y e sint y"-y"+y-y= g(1). In each of Problems 5 and 6, find the general solution of the given differential equation. Leave your answer in terms of one or more integrals. 11. Find a formula involving integrals for a particular solution of th differential equation es: Reducti Undeter y4) - y= g(1). 5. y-y+y-y-secr, -<1< 2 Hint: The functions sin t, cost, sinh t, and cosh set of solutions of the homogeneous equation. Mechan form a fundamen 6. y"-y= csc t, 0<t<T with Co In each of Problems 7 and 8, find the solution of the given initial-value problem. Then plot a graph of the solution. @ 7. y"-y"+y'-y sect: y(0) = 2, y'(0) = -1, y'(0) = 1 12. Find a formula involving integrals for a particular solution of differential equation Solutior %3D y"-3y" +3y-y g(1). as: 3. (:)). (4) -- (4)-- If g(t) =1e, determine Y(t). = 1, = 2, y' @ 8. y"-y' = tan r; 12 (:)-) ns:

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Expert Answer

Step 1

 In the method of variation of method, the complete solution is calculated just by the CF, NO PI is needed.

Step 2

For the differential equation, y’’’y’’+y’-y=e-tsint the characteristic equation is,

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m – m² + m –1= 0 m² (m–1)+1(m –1) = 0 (m² +1)(m – 1) = 0 m = 1, ±i

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Step 3

For m=1,i, -i the yc(t)...

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y.(1) =c,e"" +c, cost + c, sint = c;e' +c, cost +c; sint

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