(1) Verify that the indicated function is a solution of given DE: (i) y" – 4y' + 4y = 0; y = c1e²x + c2xe²x (ii) y"+y = tan x; y = -(cos x) In(sec x + tan x)

Q: a3 2- Find the third derivative for = sin(÷) for f(x, y) ax3 -2sin?cos()+2 sin A:- ) sin()+3cos )…

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Q: 1. Find y'. y = x² In(tan-1(2x))

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Q: If f(x) = cos x – 7 tan æ, then f'(z) -sin( x ) – 7 sec²(x) = f'(2) =

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Q: Show that there is exactly one solution f E L'([0, 1]) of the equation f(x) = 7 + sin(x) %3D

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Q: Let f(x) = Then f'(x) = π The equation of the tangent line to y = f(x) at a = can be written in the…

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Q: Find f. f"(0) = sin(0) + cos(0), f(0) = 3, f'(0) = 4 f(0) =

A: Objective is to find f.

Q: x cos'(y)dx + tan y dy = 0, y(4) = 7 2. X COS

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Q: (e) Find y' and y" for y = sin (x+y). %3D (f) Find y' for sin x+ sin y = 1. %3D (g) Find y' for x…

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Q: Question: Find the derivative. tan 3x y = (x + 7)ª Select one: 3 (x + 7) sec?r (3x) – 4 tan (3x) y'…

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Q: 2. Solve 3 y'(z) + y(x) = y²(z) (cosz - sinx)

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Q: #15) Determine y' a.y=sin 2x b. y = x² sinx co y = sin(-x) Determine #16.> d.) y = cos x sin x e-> y…

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Q: For cos2(xy) =, evaluate y' when x = 4

A: Given that: cos2(xy)=14

Q: -1 Find the derivative of the function: y=6 sin ¹¹ (x²). y'= y'= O y'= 12x y'= -X 2x 2 + X 12x 4 4 X…

A: Given y =6 sin-1 (x2 )

Q: Let f (x) = 4 tan (7x). Find the values of the coefficients in the expressions for f' (x) and f"(x)…

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Q: 5. Let F(x) : = a sin(3.x) + bx cos(3x). Find constants a, b such that F'(x) = 2x sin(3x) and verify…

A: The detailed solution is given below Thank you

Q: a2f Solve the second derivative forf (x,y,z) = e 2+y²) əx2 sin (x² + sin (x² +y²' sin -2e x2 sin…

A: We have to find the second derivative ∂2f∂x2 for fx,y,z=esinx2+y2z.

Q: f2(x) = (4x + 5)°, fa(x)= (2x² +x)VI, fa(x) cos(3r) – 2 sin? x, COS

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Q: (2) If y = (sin-1(x))², prove that (1- x2)y" – xy' - 2 = 0.

A: here first we will derivative with respect to x then squaring both side and put value of the y then…

Q: It is given that f(x) = tan (1 +x). %3D Find f(0) and f '(0), and show that f "(0) = -

A: We know thatddxtan-1x=11+x2

Q: Let f (x) = 2 tan (5x). Find the values of the coefficients in the expressions for f'(x) and f " (x)…

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Q: Find the derivative with respect to x. (c) y = v7x – x4 (d) y = (1+ sin 7.x)-5 (a) y = (5x – 2)*…

A: This question is related to derivative, we will solve it using given information.

Q: Find y' where y = cos (ze19 =) – gec = Cos ( xe-

A: on doing differentiation with respect to x we get option C

Q: 21. y = csc 2x 22. y = tan 2 Vx* + 1 x tan x 24. y 23. у x? + 1 1 + sin? x

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Q: I ind the second derivative(y") of the following functions and simplify the final results. 1. y = x…

A: Differentiate using the rules of derivative.

Q: _et f(x) = Then f'(x) = 6 sin 6 sin x + 4 cos π The equation of the tangent line to y = f(x) at a =…

A: fx=6sinx6sinx+4cosx

Q: 2. Find the derivatives of the following functions. (i) y = In²(cos ¹(2x))+ √coth x² (ii) y =…

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Q: (1) C) What is y'(x) for a function defined by the equation x sin(ay) = y+ e" as it passes through a…

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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.