2. (a) Evaluate [ x* V1+4x² dx. (b) Evaluate 8x° +13x dx. (x² +2)²

Q: 3- y = sec x prove that d? y + y = 2y3 dx2

A: We have to prove the given differential equation to be true.

Q: d²y If y=cos 4xr, then at x = 0 dx?

A: Given,     y=cos4x

Q: -3 Evaluate: ²¹²² ²³ y cos(z³) dx dy dz

A:

Q: 38. A. -arcsin x x Vx2 – a² 39. 1 В. -arccos x a2 + x2 40. 1 С. | -arccot x Va? – x2 - D. -arccs x…

A: Match the following

Q: dx- B- If y = cos*x – sin*x , prove that: - d²y = -4cos2x dx²

A:

Q: 4 dx. x'V16-x (c) By using x= 4sin (0), find

A: We have to find integration by substitution

Q: ential equati x²y dy = C

A:

Q: af dy of f(x, y) = tan (x²y³ + 2xy)? What is the

A: Given:     To determine:

Q: 12. Use the substitution x = sec 0, where 0 < 0 < T/2 or T < 0 < 3#/2, to evaluate Vx2 1 dx x*

A: We will evaluate the given integral by using substitution method as shown below

Q: b) Z = Arctan ( x² /y³)

A:

Q: 2 4 Differentiate cxprersion (use D for dy X y' w. respect tox

A: on diffrentiating with respect to x we get

Q: Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8); fi-) U =

A: we can solve this as follows:

Q: y = (sin x +cos x) sec x = sec² x

A:

Q: Find the solution and the interval of existence for: (1-x2)dy/dx + 2y = (1-x)2

A:

Q: Consider - 2x 2) (² f(x, y) = e fxx + fyy = 0 cos 2y - e -2y cos 2x. Verify that

A:

Q: fo (9 cos(x) – 2 sec²(x)) dx

A:

Q: RFind dy/dx in terms of x and y if cos²(8y) + sin²(8y) = y+4. |

A: Given query is to find the derivative of the given expression. Pp

Q: 4x + Vx Y = (a) 7x2 – 3 et (b) y = 7x³ cos x 19

A:

Q: d²y dy and dx Find if x = sin 3z and y = tan23z dx2

A: It can be observed that both the variables x and y are the functions of  z, therefore to find the…

Q: V sin2(x32 - 1) dx, with u x32 - 1

A: Consider the integral ∫(√x)sin2 (x3/2 - 1) dx.

Q: 2) Integrate using trigonometric substitution (a) s dx 4- x² (b) S, dx Vx2- 9

A:

Q: b) Calculate dx for n = 6, using Trapezoidal rule. 1+x5

A: Trapezoidal rule:   where

Q: If f = 5xy2 - X³Y then find df at (1,1) dy %3D

A:

Q: Q3) A: Find y if: a- y = sin 2x + 2 cos x c- y = sin √x (As... b- y = secx. csc x d- y = tan(2x + 1)

A: Please see the attachment

Q: 2. If y = e-* cos 2x, then f'(x) = %3D

A: Given : fx = e-xcos2x

Q: f [Mesin(2x) dx = e$in(2x) + C, then the M is equal to- If

A:

Q: 1. Use a substitution to find (6a2 + 2)e*+# dx showing working.

A:

Q: Finf y' if x=Arccos(1-t), y=/2t-t2.

A: We can find the answer using chain rule as below.

Q: Q NO3)IF y=Jsinx + Vsinx + Vsinx + . then show that(2y-1) = cosx .And Also Find dx (5vcosx) dx

A: Our objective is to find the derivative of above functions.

Q: cos(x) dæ with n = 4 is: The Trapezoidal Rule approximation for

A: Given ∫14 cos(x)x dx   n= 4 here f(x) =cos(x)x

Q: 2- is the D.E:(3x² + y cosx )dx + ( sinx-4y )dy = 0 | a- - Exact b- linear D.E (show that)

A:

Q: ution of y" + 4y = cos 2x b

A:

Q: 3+6 b. (2xy-sin x) dx + (x-Cos yldy

A: Topic:- solution of differential equations

Q: 1. Evaluate dx sec2 x (a) Ja(x+ 4) (b) A. dx V1- tan?x VT(x + 4)

A:

Q: Solve by finding an I.F 1. xdy – ydx = (x² + y² }£x y - sin x 2. dy + -dx = 0 ......

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: tan (e) 1+z* (d) (22+ cosh 2z)dz () z coshz dz zp-

A:

Q: (4D3 + D)y = e2* A. y = c1 + c2 cos 0.5x + c3 sin 0.5x +e2x 33 B. y = c1 + c2 cos 2x + c3 sin 2x…

A: From the given problem: 4D3+Dy=e2x The auxiliary equation is: 4m3+m=0m4m2+1=0m=0  or…

Q: 1- find dy/dx as a function of x. y = sec (tan x)

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: dy at x=0. dx If y = (3x² + 4x + 5)(e* – cos 2x), then find %3D -

A:

Q: d (cos² 2x)=-2sin 4.x dx 12 Show that %3D

A:

Q: Q2) a) Find an® sec²y dy b) Is cos4xv2-stn4x dx = (2- sin4x) +Ctrue or false? show me

A: Use fundamental theorem of calculus for part (a)  For part (b) use substitution method

Q: a) f(x) = 10r23 + el2 + 53 - In(r)+ arccos(x) %3D

A: Given

Q: 1. Use residues to evaluate 1 : xe dx , (a>0).

A: Let fz=ze-zz2+a2 The poles of the function fz are ±ai Choose the contour C consisting of the real…

Q: 3 T - 2 Evaluate: J. Soves y cosx³dy dx

A:

Q: What is the I.F of (x^2y^3+2y)dx+(2x-2x^3y^2)dy=0?

A: Find the integrating factor

Q: (b)/ Use the Trapezoidal Rule with n = 4 to estimate ſ,(x3 + 1)dx.

A: To find: ∫-11x3 + 1dx using trapezoidal rule with n=4

Q: Solve for x = u + 1 and y = 2u² - 4u d²y dx² with respect to the parameter u

A:

Q: Evaluate the dy/dx> d) y=e-x((cos(2x)) a) -e-x(cos2x+2sin2x) b) e-x(sin2x-cos2x) c)…

A: y=e-xcos(2x)differentiate…

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.