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Find constants a and b which make the function x(t)=acos(t)+bsin(t) a solution to the differential equation d2x/dt2 - dx/dt = cos(t)

Question

Find constants a and b which make the function x(t)=acos(t)+bsin(t) a solution to the differential equation d2x/dt- dx/dt = cos(t)

check_circleAnswer
Step 1

If x(t)=a cos (t) + b sin(t) is a solution to the differential equation d2x/dt- dx/dt = cos(t) then x(t) = a cos(t) + b sin(t) satisfied the given differential equation.

So, differentiate x(t) = a cos(t) + b sin(t) with respect to the variable (t).

x)acos ()b sin(t)
d
dx
(cos (t))bsint
dt
dt
dt
dx
- a(-sin ())b(cos ())
dt
dx
a sin(t)+b cos (t)
dt
help_outline

Image Transcriptionclose

x)acos ()b sin(t) d dx (cos (t))bsint dt dt dt dx - a(-sin ())b(cos ()) dt dx a sin(t)+b cos (t) dt

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Step 2

Again, differentiate with respect to t:

 

dx
=-a sin(t)+b cos (t)
dt
d2x
(sin(t))bcos (t))
=-a
dt2
dt
dt
d2x
-a cos(t)+b(-sin ())
dt2
d2x
-a cos (t)-b(sin ())
dt
help_outline

Image Transcriptionclose

dx =-a sin(t)+b cos (t) dt d2x (sin(t))bcos (t)) =-a dt2 dt dt d2x -a cos(t)+b(-sin ()) dt2 d2x -a cos (t)-b(sin ()) dt

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Step 3

Substitute the expression of dx/dt and d2x/dt2 

 

In the equation d2x/dt2&...

d2x dx
-cos()
dt
dt2
-a cos(t)-b(sin (t))-(-a sin(t)+b cos(t))= cos (t)
(-a-b) cos (t)(ba) sin (t)= cos()
COs
help_outline

Image Transcriptionclose

d2x dx -cos() dt dt2 -a cos(t)-b(sin (t))-(-a sin(t)+b cos(t))= cos (t) (-a-b) cos (t)(ba) sin (t)= cos() COs

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