inverse oflaplace (3s^2)/((s^2+4)^2)

{ "query": { "display": "inverse laplace $$\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}$$", "symbolab_question": "LAPLACE#inverselaplace \\frac{3s^{2}}{(s^{2}+4)^{2}}" }, "solution": { "level": "PERFORMED", "subject": "Calculus", "topic": "Laplace", "subTopic": "Inverse", "default": "\\frac{3\\sin(2t)+6t\\cos(2t)}{4}" }, "steps": { "type": "interim", "title": "Laplace Inverse Transform of $$\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}:{\\quad}\\frac{3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)}{4}$$", "input": "L^{-1}\\left\\{\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}\\right\\}", "steps": [ { "type": "interim", "title": "Take the partial fraction of $$\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}:{\\quad}\\frac{3}{s^{2}+4}-\\frac{12}{\\left(s^{2}+4\\right)^{2}}$$", "input": "\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}", "steps": [ { "type": "interim", "title": "Create the partial fraction template using the denominator $$\\left(s^{2}+4\\right)^{2}$$", "result": "\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}=\\frac{a_{1}s+a_{0}}{s^{2}+4}+\\frac{a_{3}s+a_{2}}{\\left(s^{2}+4\\right)^{2}}", "steps": [ { "type": "step", "primary": "For $$\\left(s^{2}+4\\right)^{2}\\:$$add the partial fraction(s): $$\\frac{a_{1}s+a_{0}}{s^{2}+4}+\\frac{a_{3}s+a_{2}}{\\left(s^{2}+4\\right)^{2}}$$" }, { "type": "step", "result": "\\frac{3s^{2}}{\\left(s^{2}+4\\right)^{2}}=\\frac{a_{1}s+a_{0}}{s^{2}+4}+\\frac{a_{3}s+a_{2}}{\\left(s^{2}+4\\right)^{2}}" } ], "meta": { "interimType": "Partial Fraction Templates Top 1Eq" } }, { "type": "step", "primary": "Multiply equation by the denominator", "result": "\\frac{3s^{2}\\left(s^{2}+4\\right)^{2}}{\\left(s^{2}+4\\right)^{2}}=\\frac{\\left(a_{1}s+a_{0}\\right)\\left(s^{2}+4\\right)^{2}}{s^{2}+4}+\\frac{\\left(a_{3}s+a_{2}\\right)\\left(s^{2}+4\\right)^{2}}{\\left(s^{2}+4\\right)^{2}}" }, { "type": "step", "primary": "Simplify", "result": "3s^{2}=\\left(a_{1}s+a_{0}\\right)\\left(s^{2}+4\\right)+a_{3}s+a_{2}" }, { "type": "step", "primary": "Expand", "result": "3s^{2}=a_{1}s^{3}+4a_{1}s+a_{0}s^{2}+4a_{0}+a_{3}s+a_{2}" }, { "type": "step", "primary": "Group elements according to powers of $$s$$", "result": "3s^{2}=a_{1}s^{3}+a_{0}s^{2}+s\\left(4a_{1}+a_{3}\\right)+\\left(4a_{0}+a_{2}\\right)" }, { "type": "step", "primary": "Equate the coefficients of similar terms on both sides to create a list of equations", "result": "\\begin{bmatrix}4a_{0}+a_{2}=0\\\\4a_{1}+a_{3}=0\\\\a_{0}=3\\\\a_{1}=0\\end{bmatrix}" }, { "type": "interim", "title": "Solve system of equations:$${\\quad}a_{0}=3,\\:a_{1}=0,\\:a_{2}=-12,\\:a_{3}=0$$", "result": "a_{0}=3,\\:a_{1}=0,\\:a_{2}=-12,\\:a_{3}=0", "steps": [ { "type": "step", "result": "\\begin{bmatrix}4a_{0}+a_{2}=0\\\\4a_{1}+a_{3}=0\\\\a_{0}=3\\\\a_{1}=0\\end{bmatrix}" }, { "type": "step", "primary": "Substitute $$a_{0}=3$$", "result": "\\begin{bmatrix}a_{1}=0\\\\4\\cdot\\:3+a_{2}=0\\\\4a_{1}+a_{3}=0\\end{bmatrix}" }, { "type": "interim", "title": "Simplify", "input": "4\\cdot\\:3+a_{2}=0", "steps": [ { "type": "interim", "title": "Simplify $$4\\cdot\\:3+a_{2}:{\\quad}12+a_{2}$$", "input": "4\\cdot\\:3+a_{2}", "steps": [ { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:3=12$$", "result": "=12+a_{2}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "12+a_{2}=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\begin{bmatrix}a_{1}=0\\\\12+a_{2}=0\\\\4a_{1}+a_{3}=0\\end{bmatrix}" }, { "type": "step", "primary": "Substitute $$a_{1}=0$$", "result": "\\begin{bmatrix}12+a_{2}=0\\\\4\\cdot\\:0+a_{3}=0\\end{bmatrix}" }, { "type": "interim", "title": "Simplify", "input": "4\\cdot\\:0+a_{3}=0", "steps": [ { "type": "interim", "title": "Simplify $$4\\cdot\\:0+a_{3}:{\\quad}a_{3}$$", "input": "4\\cdot\\:0+a_{3}", "steps": [ { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0+a_{3}" }, { "type": "step", "primary": "$$0+a_{3}=a_{3}$$", "result": "=a_{3}" } ], "meta": { "solvingClass": "Solver", "interimType": "Algebraic Manipulation Simplify Title 1Eq" } }, { "type": "step", "result": "a_{3}=0" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Title 0Eq" } }, { "type": "step", "result": "\\begin{bmatrix}12+a_{2}=0\\\\a_{3}=0\\end{bmatrix}" }, { "type": "step", "result": "\\begin{bmatrix}12+a_{2}=0\\end{bmatrix}" }, { "type": "interim", "title": "Isolate $$a_{2}\\:$$for $$12+a_{2}=0:{\\quad}a_{2}=-12$$", "input": "12+a_{2}=0", "steps": [ { "type": "interim", "title": "Move $$12\\:$$to the right side", "input": "12+a_{2}=0", "result": "a_{2}=-12", "steps": [ { "type": "step", "primary": "Subtract $$12$$ from both sides", "result": "12+a_{2}-12=0-12" }, { "type": "step", "primary": "Simplify", "result": "a_{2}=-12" } ], "meta": { "interimType": "Move to the Right Title 1Eq", "gptData": "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" } } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Isolate Title 2Eq" } }, { "type": "step", "primary": "The solutions to the system of equations are:", "result": "a_{0}=3,\\:a_{1}=0,\\:a_{2}=-12,\\:a_{3}=0" } ], "meta": { "solvingClass": "System of Equations", "interimType": "Partial Fraction Solve System Equation 0Eq" } }, { "type": "step", "primary": "Plug the solutions to the partial fraction parameters to obtain the final result", "result": "\\frac{0\\cdot\\:s+3}{s^{2}+4}+\\frac{0\\cdot\\:s+\\left(-12\\right)}{\\left(s^{2}+4\\right)^{2}}" }, { "type": "step", "primary": "Simplify", "result": "\\frac{3}{s^{2}+4}-\\frac{12}{\\left(s^{2}+4\\right)^{2}}" } ], "meta": { "solvingClass": "Partial Fractions", "interimType": "Algebraic Manipulation Partial Fraction Top Title 1Eq" } }, { "type": "step", "result": "=L^{-1}\\left\\{\\frac{3}{s^{2}+4}-\\frac{12}{\\left(s^{2}+4\\right)^{2}}\\right\\}" }, { "type": "step", "primary": "Use the linearity property of Inverse Laplace Transform:<br/>For functions $$f\\left(s\\right),\\:g\\left(s\\right)$$ and constants $$a,\\:b:{\\quad}L^{-1}\\{a{\\cdot}f\\left(s\\right)+b{\\cdot}g\\left(s\\right)\\}=a{\\cdot}L^{-1}\\{f\\left(s\\right)\\}+b{\\cdot}L^{-1}\\{g\\left(s\\right)\\}$$", "result": "=L^{-1}\\left\\{\\frac{3}{s^{2}+4}\\right\\}-L^{-1}\\left\\{\\frac{12}{\\left(s^{2}+4\\right)^{2}}\\right\\}" }, { "type": "interim", "title": "$$L^{-1}\\left\\{\\frac{3}{s^{2}+4}\\right\\}:{\\quad}\\frac{3}{2}\\sin\\left(2t\\right)$$", "input": "L^{-1}\\left\\{\\frac{3}{s^{2}+4}\\right\\}", "steps": [ { "type": "step", "result": "=L^{-1}\\left\\{\\frac{3}{2}\\cdot\\:\\frac{2}{s^{2}+2^{2}}\\right\\}" }, { "type": "step", "primary": "Use the constant multiplication property of Inverse Laplace Transform:<br/>For function $$f\\left(t\\right)$$ and constant $$a:{\\quad}L^{-1}\\{a{\\cdot}f\\left(t\\right)\\}=a{\\cdot}L^{-1}\\{f\\left(t\\right)\\}$$", "result": "=\\frac{3}{2}L^{-1}\\left\\{\\frac{2}{s^{2}+2^{2}}\\right\\}" }, { "type": "step", "primary": "Use Inverse Laplace Transform table: $$L^{-1}\\{\\frac{a}{s^{2}+a^{2}}\\}=\\sin\\left(at\\right)$$", "secondary": [ "$$L^{-1}\\{\\frac{2}{s^{2}+2^{2}}\\}=\\sin\\left(2t\\right)$$" ], "result": "=\\frac{3}{2}\\sin\\left(2t\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$L^{-1}\\left\\{\\frac{12}{\\left(s^{2}+4\\right)^{2}}\\right\\}:{\\quad}\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)$$", "input": "L^{-1}\\left\\{\\frac{12}{\\left(s^{2}+4\\right)^{2}}\\right\\}", "steps": [ { "type": "step", "result": "=L^{-1}\\left\\{\\frac{3}{4}\\cdot\\:\\frac{2\\cdot\\:2^{3}}{\\left(s^{2}+2^{2}\\right)^{2}}\\right\\}" }, { "type": "step", "primary": "Use the constant multiplication property of Inverse Laplace Transform:<br/>For function $$f\\left(t\\right)$$ and constant $$a:{\\quad}L^{-1}\\{a{\\cdot}f\\left(t\\right)\\}=a{\\cdot}L^{-1}\\{f\\left(t\\right)\\}$$", "result": "=\\frac{3}{4}L^{-1}\\left\\{\\frac{2\\cdot\\:2^{3}}{\\left(s^{2}+2^{2}\\right)^{2}}\\right\\}" }, { "type": "step", "primary": "Use Inverse Laplace Transform table: $$L^{-1}\\{\\frac{2a^{3}}{\\left(s^{2}+a^{2}\\right)^{2}}\\}=\\sin\\left(at\\right)-at\\cos\\left(at\\right)$$", "secondary": [ "$$L^{-1}\\{\\frac{2\\cdot\\:2^{3}}{\\left(s^{2}+2^{2}\\right)^{2}}\\}=\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)$$" ], "result": "=\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\frac{3}{2}\\sin\\left(2t\\right)-\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)" }, { "type": "interim", "title": "Simplify $$\\frac{3}{2}\\sin\\left(2t\\right)-\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right):{\\quad}\\frac{3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)}{4}$$", "input": "\\frac{3}{2}\\sin\\left(2t\\right)-\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)", "result": "=\\frac{3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)}{4}", "steps": [ { "type": "interim", "title": "Multiply $$\\frac{3}{2}\\sin\\left(2t\\right)\\::{\\quad}\\frac{3\\sin\\left(2t\\right)}{2}$$", "input": "\\frac{3}{2}\\sin\\left(2t\\right)", "result": "=\\frac{3\\sin\\left(2t\\right)}{2}-\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\sin\\left(2t\\right)}{2}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Multiply $$\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)\\::{\\quad}\\frac{3\\left(-2t\\cos\\left(2t\\right)+\\sin\\left(2t\\right)\\right)}{4}$$", "input": "\\frac{3}{4}\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)", "result": "=\\frac{3\\sin\\left(2t\\right)}{2}-\\frac{3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)}{4}", "steps": [ { "type": "step", "primary": "Multiply fractions: $$a\\cdot\\frac{b}{c}=\\frac{a\\:\\cdot\\:b}{c}$$", "result": "=\\frac{3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)}{4}" } ], "meta": { "interimType": "Generic Multiply Title 1Eq" } }, { "type": "interim", "title": "Least Common Multiplier of $$2,\\:4:{\\quad}4$$", "input": "2,\\:4", "steps": [ { "type": "definition", "title": "Least Common Multiplier (LCM)", "text": "The LCM of $$a,\\:b$$ is the smallest positive number that is divisible by both $$a$$ and $$b$$" }, { "type": "interim", "title": "Prime factorization of $$2:{\\quad}2$$", "input": "2", "steps": [ { "type": "step", "primary": "$$2$$ is a prime number, therefore no factorization is possible", "result": "=2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRl8ZboA8wPLg0yhI4RzfjFw/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp1+G9v2aKasChgV65VW8cTW" } }, { "type": "interim", "title": "Prime factorization of $$4:{\\quad}2\\cdot\\:2$$", "input": "4", "steps": [ { "type": "step", "primary": "$$4\\:$$divides by $$2\\quad\\:4=2\\cdot\\:2$$", "result": "=2\\cdot\\:2" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRsG/uC0ndYtZpJL4uAxK7FI/y9DKGIPglJ+qMi9xDu2KE1OovxZAaXg7BtrFPk4UcCzRnGgMN6CYRfod7Mq0dp39fF/zAtU5baHQ1hwgXA+n" } }, { "type": "step", "primary": "Multiply each factor the greatest number of times it occurs in either $$2$$ or $$4$$", "result": "=2\\cdot\\:2" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:2=4$$", "result": "=4" } ], "meta": { "solvingClass": "LCM", "interimType": "LCM Top 1Eq" } }, { "type": "interim", "title": "Adjust Fractions based on the LCM", "steps": [ { "type": "step", "primary": "Multiply each numerator by the same amount needed to multiply its<br/>corresponding denominator to turn it into the LCM $$4$$" }, { "type": "step", "primary": "For $$\\frac{3\\sin\\left(2t\\right)}{2}:\\:$$multiply the denominator and numerator by $$2$$", "result": "\\frac{3\\sin\\left(2t\\right)}{2}=\\frac{3\\sin\\left(2t\\right)\\cdot\\:2}{2\\cdot\\:2}=\\frac{6\\sin\\left(2t\\right)}{4}" } ], "meta": { "interimType": "LCD Adjust Fractions 1Eq" } }, { "type": "step", "result": "=\\frac{6\\sin\\left(2t\\right)}{4}-\\frac{3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)}{4}" }, { "type": "step", "primary": "Since the denominators are equal, combine the fractions: $$\\frac{a}{c}\\pm\\frac{b}{c}=\\frac{a\\pm\\:b}{c}$$", "result": "=\\frac{6\\sin\\left(2t\\right)-3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)}{4}" }, { "type": "interim", "title": "Expand $$6\\sin\\left(2t\\right)-3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right):{\\quad}3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)$$", "input": "6\\sin\\left(2t\\right)-3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)", "result": "=\\frac{3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)}{4}", "steps": [ { "type": "interim", "title": "Expand $$-3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right):{\\quad}-3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)$$", "input": "-3\\left(\\sin\\left(2t\\right)-2t\\cos\\left(2t\\right)\\right)", "result": "=6\\sin\\left(2t\\right)-3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)", "steps": [ { "type": "step", "primary": "Apply the distributive law: $$a\\left(b-c\\right)=ab-ac$$", "secondary": [ "$$a=-3,\\:b=\\sin\\left(2t\\right),\\:c=2t\\cos\\left(2t\\right)$$" ], "result": "=-3\\sin\\left(2t\\right)-\\left(-3\\right)\\cdot\\:2t\\cos\\left(2t\\right)", "meta": { "practiceLink": "/practice/expansion-practice", "practiceTopic": "Expand Rules" } }, { "type": "step", "primary": "Apply minus-plus rules", "secondary": [ "$$-\\left(-a\\right)=a$$" ], "result": "=-3\\sin\\left(2t\\right)+3\\cdot\\:2t\\cos\\left(2t\\right)" }, { "type": "step", "primary": "Multiply the numbers: $$3\\cdot\\:2=6$$", "result": "=-3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7MWbLmQ2yn0EIGu1sE/yIIbcYamIWEuBNU/KTkj8uFWvdd47a0hQ8flDbGsI5To1d90acrR+rqVDKu/7m7mCNoeN+q42ekh7iALRgSPFoOwNkS3dlcCKpQTQcheuut7MkVOh9J0vvvJA6j+t9n7lRz3jrGc4YP9Mxa7gMPV63BH3+1RM4fDhQO2vTe6+S6ui4" } }, { "type": "step", "primary": "Add similar elements: $$6\\sin\\left(2t\\right)-3\\sin\\left(2t\\right)=3\\sin\\left(2t\\right)$$", "result": "=3\\sin\\left(2t\\right)+6t\\cos\\left(2t\\right)" } ], "meta": { "interimType": "Algebraic Manipulation Expand Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7RIOi4PDLpvhBkkrsE3AA0aXh6WG7fQn6WAqzg296O206YOTojanwrUzBi5jksheucJChiVhDxT5N/LHSTLMjyLZZxxagPdGvCRMpLWfFPAWM/r2fecxnpmz8OvkHj0aGeqXxdc+rps1CUyb7fqI2GTVIiTVjdAWrP2YpqywGxnz9Eeki8jCf4XPlNPZcM1F3WCldfFuuYzR4R5dc+mLlRw==" } } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7hbqZttgXqBCr9XmUErpy5bBWTr/alYmSjGS/n1YGtK+y4Ywrrmd9IfdSqphqiUXHtxhqYhYS4E1T8pOSPy4Va913jtrSFDx+UNsawjlOjV2t8xE/KctJ8/dVh7U5CV1E02jdFOQRPh97/i9WL4+TaFKYK1mos4N/UhDm9MZkK/TWwPs1+Gw97t4MeuaNjSYTwPBBw3byB5RPtpSy6YLt2geO8mN9vvrPNe5bTmG3OE/6qIUxxL2JGsAA93XmiFCYpeHpYbt9CfpYCrODb3o7bT3C22wQcG4cUE2AcmfIScY=" } } ], "meta": { "solvingClass": "Laplace" } } }