x + y3 -1 Еxample If u = tan prove that x- y ôu ди (i) x. + y. sin 2u

Q: (a) cos(iz) = cos(iz) (b) sin(iz) = sin (iz) for all z; if and only if z = nлi (n = 0, +1, +2, ...).

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Q: State Leibnitz's theorem. If V1- x² y =sinx, then prove that (1-x) yn+2– (2n+1) xyn+1=n°yn= 0.

A: The Leibnitz theorem is used to generalize the product rule of differentiation. It states that if…

Q: d -1 ) Prove that cos 1(x) : dx V1-1

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Q: 13. To evaluate f 3cos*x-1 dx. the integrand can be transformed to 1-sin2x a. [(3cos x+ csc²x)dx b.…

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Q: (x²+y² 4. If u = sin-1 ( ) then prove that by Euler's theorem, x+Vy. ди X + y: дх ди 3 tan u. 2

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Q: 1. If y= tan x, prove that y"= 2(y² +1)(3y² +1).

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Q: ). y² = 4x, 2x - y = 12 ). y = In x³, y = In x, x = e ):y = 2sin x, y = sin x from x = – to x : %3D…

A: As per bartleby guidelines for more than one part only first part is to be answered , please upload…

Q: 1. If y= tan x, prove that y``= 2(y² +1)(3y² +1).

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Q: ди k > 0 at Ans. u = e'(A,eka! cosh ax + Bjekai sinh œx) u = e¯'(A2e' u = e"(Azx + B3) k ¯kai 'cos…

A: The given problem is to find the solution for the given partial differential equation in different…

Q: sin(nz) b) E: on (0, 2x] n-1 AcoAnA

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Q: (g) z-1 1+! (h)) u = 1 z +1 (i) y= cos vsin(tan Tx) (j) tan(x/y) = x + y %3D

A: As per our guidelines we are supposed to solve only first three subparts of your first problem. For…

Q: 3. u, = u, - sin x cos t, 1 >0 В.С.: и(0, г) %3D0, = () 1.С.: «(х, 0) %3D 0, u,(x. 0) = 0

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Q: 3] y = 12x - 2x+ 5 Degree: Sign of LC: as x→-0, y→. as x 0, y -

A: We have to find the degree and sign of LC and find endpoints.

Q: (x-2) (g-9) 2- 3X+uX- %3D 29-u | Sind general Salantion

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Q: 4 sin(x) cos(z) + 2 sin(z) – 2cos(z) – 1 = 0

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Q: At what points (x, y) in the plane are the functions in Exercises 31–33 continuous?

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Q: Cos z Residue of cos. at z = 0 is %3D 1. 1 -1 3. 2 4. 0 2.

A: The residue of a function: f(z) at z=a is defined as Resz=af(z)=limz→az-af(z).

Q: 3. Show that (i) r(x)T (-x) =- (ii) r T sec TX x sin TX

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Q: у" + 25у %3D 0, у(0) 3 6, у'(0) 3D —4 Y(x) = | 5 cos( 5x) – sin(5x) sin (5x) x

A: We have given differential equation y"+25y=0 with initial conditions y(0) =6, y'(0) =-4 We have to…

Q: 4  find The indefinite integral

A: To evaluate the given integral

Q: (x²+y2 ) prove ne du + y ax -2 sin2u ( Euler's Rule) 5. If u = cot-1 that x ду x+y

A: We use the Euler's rule of partial differential in homogeneous function to solve this problem.

Q: Hi, there are two questions a) and b) attached  Thanks so Much!!!!

A: (a)consider the given complex number:

Q: 5. y + 4y = sin x + 3 cos x Ans: y = C₁ + C₂ cos 2x + C3 sin 2x + sinx-cosx

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Q: iven po r (t) = (sin (t), cos (t), In(cos (t))), (0, 1,0). T = 1 1 1 - N = V2 1 1 B = V2

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Q: ♥ дzu ахбу = сезх созу, Ans: u(x,y) = sinx siny + x² + 2 cosy - #2² X 1 бу исx;#) =X бx U(x,п) =X,…

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Q: (1) y = In(x+ 4)- x tan (2) xy + xy x + 3y (3) Iny e' sin

A: In this question we will use basic of calculus i.e concept of differentiation. We will simply…

Q: Q.3 Use ODE methods to solve the BVP: U = 18xy + sin(2x- y) U(0, y) = y -cos y, U(x,0) = x² -cos2x,…

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Q: a. Show that If z=x+ iy where y 1 b. Evaluate: cosh (iz) + sin (2)

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Q: ered # tion Bi Q) choose one only 1. If (x + y)² = x² + y² prove that = - dx 2. If y = sin³ () cos³…

A: To find the differentiation of given functions.

Q: 9. y= sin (r - (z+ 10. y = sin 4 11. y = cos (x – 4. (z + 12. y = coS 13. y = 1+ sin x 14. y = -1+…

A: Matching the graph of functions given by questions 10, 12 and 14

Q: 26. y'+y tan x = cos x; y(x) = (x + C) cos x, y(n) = 0 %3D 25. y' = 3x²(y² + 1); y(x) = tan(x³ + C),…

A: 25)y'=3x2y2+1---Iyx=x3+Cy0=0solution:yx=x3+C⇒y'x=sec2x3+Cddxx3+C=sec2x3+C3x2y'x=3x2sec2x3+C---Aputti…

Q: 3Tt of cSC X du If u = f(x), then dx cos X . Compute %3D 2 sin x и -

A: Given ,  u = f(x)

Q: 7. Prove: (a) sin iz = i sinh z. (b) | cos z|? = cos² x + sinh° y where, z = x + iy. COS

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Q: 9. Evaluate y' of y = e-2*sin3x a. 3e-cos3x- 2y b. 3e-cos3x- 3y c. 3ecos3x - y d. 3e-cos3x +y e. Not…

A: Explanation of the answer is as follows

Q: 6. sin x (1 + sin x cos x) dx y csc² ( tan (y²)) dy 7. cos²(y²) sec (11w) tan (11w) V (2 – sec…

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Q: (a) If u(x, y) = cos-1*" + y* ) x² + y? = COS then prove that ди du + y = -2cotu

A: Given: ux,y=cos-1x4+y4x2+y2

Q: sin z f(2) = cosz Bring your answer to the simplest possible form.

A: Find derivative 4f as per student required

Q: 5. y + 4y= sin x + 3 cos x Ans: y = C₁+C₂ cos 2x + C3 sin 2x + sin x x-co COS X

A: To solve the given differential equation.y'''+4y'=sin x+3 cos x

Q: 47. Let y = sec-x. Show that tan y = vx? – 1 if x > 1 and that tan y = -Vx? – 1 if x 0 on (0, 3)…

A: Consider the function of x and solve further-

Q: 2. 다음 함수 /의 1계 편도함수를 구하여라. (1) f(x, y) = 2x²+ 2xy² - y' (2) (x, y) = tan(x'y) (3) f(x, y) = e'cos2y…

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Q: -1 If cos x+cos y+cos z= T, prove that x +y +z +2xyz = 1

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Q: Suppose f(x, y) = x2 + y? where x = r cos 0 and y = r sin 0. Find and whenr = 1, 6 -: = 1, 0 =

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Q: sin If y = , then justify (1–x² )n. -(2n +1)xy, – n² y,-1 =0 applying Leibnitz's a) theorem.

A: Please check step 2 for the solution.

Q: 2) en forma de hélice x = 2 sen t, y = 2 cos t, z = 3t, 0<1< 2# y función de densidad p = 5.

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Q: 8) Prove: Prove that a general solution of the equation y" - y = 0 is given by y = c cosh t+cz…

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Q: 9. y"+y= sec² x Ans. y= c, sin x + c2 cos x0 + (sin x ) In(sec x+ tan x)–1

A: Let's find.

Q: 44. First verify by substitution that y (x) =x=1/2 cosx is one solution (for x > 0) of Bessel's…

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