In each of the following problems, use the method of reduction of order to find a sec- ond solution of the given differential equation. (c). t²y" + 3ty' + y=0, t>0; (d). xy" — y' + 4x³y=0, x>0; y1(t) = t−1 y₁(x) = sin (x²)

Q: Find y as a function of x if y(1) = 8, y′(1) = 6. y = x²y" — 6xy' – 8y = 0,

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Q: Q1. f(ath) - fla) = - 1hac? +2hu-3hza+Ih2+5 h3 %3D Find f'(x)

A: To find f'(X)

Q: y' : x² + y² sin x

A: We know thatExistence and uniqueness theorem: If f is continuous on an open rectanglethat contains…

Q: y" - √xy = sinx ; (TT) = 0. y' (π) = 11. V" (π) = 3

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Q: a) Suppose f' (x) = sin(x² ) and f(0) = 4. Use a graph of f' (x) to decide which is larger: (i) ƒ(0)…

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Q: 3y' –4x²y+ xy = -5 %3D

A: Since you have asked multiple questions in a single request, we would be answering only the first…

Q: Find f. f "(t) = 5e + 8 sin t, f(0) = 0, f(x) = 0 f(t) :

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Q: Consider y=(x2+1)2x a) find dy/dx b)Evaluate y'(1)

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Q: cy(1) = 4, y'′(1) = −2. y = x²y" +7xy +9y = xº,

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Q: Apply (3) to establish the formula for the Laplace transform of an integral, F(p) L|f(x)dx 1. and…

A: Given the equation, ℒy'=pℒy-y(0)  ⋯⋯(1) Let ℒf(x)=F(p) Also, let yx=∫0xfxdx Then, from equation (1),…

Q: of y = 2 tan x 1-tan²x

A: Solution is done in step 2

Q: Use the rule for recipocals : f '(x) f?(x) d 1 dx [ f(x) to find the following derivatives…

A: Given: Reciprocal Rule: ddx1f(x)=−f'(x)f2(x) Find: 1)  ddx1x2+2x+1=?2)  ddx1sinx+ex=?3)  ddxπx+1=?

Q: If given y'= f'(x), would would need to be given in order to find the +C

A: Given,          y,=f,x

Q: b.9f Fex)=x Sin lgix), where gL1)= , and g' (1)= -1, then f'(1)=2

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Q: Find the derivative of y. y = cosh x5 O -5x sinh xã O sinh xã O 5x4 sinh xã O sinh xã

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Q: Let g(x) = xe2r +1. Find g'(x). Don't simplify your answer. (а) 3. Let h(t) sec(t2). Find h"(t).…

A: (a)Given: