rakan asked in Science & MathematicsMathematics · 7 years ago

# can you show the steps of this differential equations problem?

Find laplace {f(t)} by first using a trigonometric identity. (Write your answer as a function of s.)

f(t)= sin(3t)cos(3t)

3/(s^(2)+36)

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• 7 years ago

Keep in mind that:

L[ sin(at) ] = a / (s² + a²)

There's actually a very simple way to solve this problem and that way is to use a Double-Angle Identity. We know that sin(2x) = 2sin(x)cos(x). Using the double-angle identity then, we can go ahead and say:

sin(2x) = 2sin(x)cos(x)

sin(2(3x)) = 2sin(3x)cos(3x)

So therefore:

sin(6x) = 2sin(3x)cos(3x)

And thus, we can go ahead and re-write the Laplace Transform as:

L[ sin(3t)cos(3t) ] = L[ (1/2)*sin(6t) ]

Applying the Laplace Transform of sin(at) will give us:

L[ (1/2)*sin(6t) ]

(1/2) * L[ sin(6t) ]

(1/2) * (6 / s² + (6²))

Which we can simplify down to:

(1/2) * (6 / s² + 36)

3 / (s² + 36)

• Anonymous
7 years ago

Hello ?,

I have posted a full solution to your question on a math help forum where I administrate so that I may provide an easy to read explanation using LaTeX:

http://www.mathhelpboards.com/f52/s-question-yahoo...

• 4 years ago

dA/dT=3(a million)-(A/210)(3) dA/dT=3-A/70 dA/dT=(210-A)/70 dA/(210-A)=a million/70dT -ln(210-A)=t/70 ln(210-A)= -t/70 e^(-t/70)=210-A A= 210-e^(-t/70) be beneficial you plug in preliminary condition.