Use the Laplace transform to solve the following initial...

Use the Laplace transform to solve the following initial value problem: y'' + 2y' = 0 y(0) = -1, y'(0) = 4 First, using Y for the Laplace transform of y(t), i.e., Y = L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation = 0 Now solve for Y(s) = and write the above answer in its partial fraction decomposition, Y(s) = A/(s+a) + B/(s+b) where a < b Y(s) = + Now by inverting the transform, find y(t) =

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00:01 We have a differential equation y double dash plus 10 y is equals to 0 10 y dash when that y of 0 is equals to 4 and y dash 0 is equals to 1 that we have to find solution to the ivp problem to ivp using laplace transform so we have to use laplace transform to find the answer to the question laplace transform on both side we have laplace transform of y double dash plus 10 times laplace transform of y dash is equals to 0 so if we take laplace transform of y equals to capital y of s we have to take laplace transform of y is equals to capital y on taking this generalization we get the laplace transform equation as laplace transform of y double dash as s square multiplied by y s minus s time y of 0 which is 4 is y dash 0 which is 1 plus 10 times laplace transform of y dash will which will be s times y of s minus y of 0 which is 4 so this is equals to 0 therefore we have the final equation as s square multiplied by y of s minus 4 s minus 1 plus 10 s times y of s minus 40 is equals to 0 so the final equation becomes s square plus 10 s multiplied by y of s minus 4 s minus 41 is equals to 0 so this is the first part of our question now we have to solve for the function y of s we have y of s is equals to 4 s plus 41 upon s times s plus 10 so this is the answer for the function y of s now we have to decompose it into the partial fraction formula so we can write y of s as a times s plus b times b divided by s plus 10 so y of s can be written as a s plus 10 s plus b so here it will be a s plus 10 times a plus b s divided by s multiplied by s plus 10 so we can write this as a plus b is equals to 4 and 10 times a is equals to 41 this are written as a is equals to 41 by 10 and value of b is equals to 4 minus 41 by 10 which will be minus 1 by 10 so we have the value for both the constants so y of s can be written as 41 by 10 s minus 1 by 10 times s plus 10 so this we have as a partial fraction formula now the answer to the question y is equals to laplace inverse of 41 by 10 s minus 1 by 10 times s plus 10 this becomes laplace in 41 by 10 times laplace inverse of 1 by s minus 1 by 10 times laplace inverse of 1 divided by s plus 10 on solving this we get y is equals to 41 by 10 minus 1 by 10 times e raised to the power minus 10 p...

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