Question

please solve both 1&2

Solve the following differential equations using the Laplace transform method 1. x" + 4x = t, x(0) = 0, x'(0) = 1. 2. x" + 2x' + x = t?, x(0) = 0, x'(0) = 1

Answer 1

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Solution © x'+4x = t ,x(0)=0, x'(o)=1 Using Laplace Transfrom $*X(5) – 5 ulo) – sk!(0) + 4XIS) - 2 ($%+4)x(3) = get XIS) = log in to using inverse Laplace Transform Ult) = {t{ West Heming, testing} xH) = 4( { } + 3 4 5 } (214)'= $ + simple t) ] Dusuar ♡ x" + 2x te= ť ,x10=0, x1(0)=1 Using Laplace Transform *XUS – SX(t) - x(9) + 2 + x(s) – 2 X ) + X(s) — L{1} (5²+28 + 1) X(S) = 1 + X19= 5Y&25+1) + ($2881) Using Inverse Laplace Transform Xa xlt) = ["{ pyja * (34,5% (ult) = 6ult) -ut + t - bet tet css Wherewithliltzea damit step function ar Rowan . . .