Prove that r(v) : dx, v> 0.

Q: 2. Show that the function u = p(xy) + Vry v(y/x) provides the equation = 0. dy? | dx²

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Q: 2 consider, Differentiate with respect to x', dy C」.2ス %3D de

A: From the image , two find a differential equation eliminating the constants from the given relation.…

Q: I=Jx Incx) dx, where b is a constant with %3D bフ)

A: Intregation of product of two variables is given below

Q: By algebraic substitution, the integral - dx dx is converted to: 4+x 2ydy (A y+4

A: Option a is the correction and other option does not satisfy the given conditions

Q: dy Differentiate parametrically to find dx in terms of t: x = 4t?,y= 2t3

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Q: How would you verify that the following equation is not exact?(-xysin(x)+2ycos(x))dx + 2xcos(x)dy =…

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Q: Where f(x, y) y calculate (–1, 1) towards the point dv (-1, 3)

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Q: dy If x = t - t3 and y = t2 - t, the value of at t = 2 is %3D %3D dx

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Q: Answer the following questions for the DE. (y – 2)dx + (3x - y)dy 0 %3D What type of DE is shown?

A: (y-2)dx+(3x-y)dy=0  is first order non  linear ordinary  differential equation

Q: dz 2. Given z = 4x2 + y²,x = sin t, y = t², find- %3D dt

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Q: f (x, y, 2) = In (x² + y? + ²) ve df x = cost, y = sint, z = tise dt It%=D1

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Q: d'y Calculate y = eX + e* dy %3D dx2

A: It is known that if  we have a function f(ax) then the derivative of the function is,   Here it is…

Q: u + c. Evaluate So (2x- y)3e dx dy with y = v, x= u-

A: To evaluate ∫02∫y22+y22x−y3eydxdy with y=v, x=u+v2.

Q: zl*dz, where C is x(t) t2, y(t) , t:2 1 Evaluate /=

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: dy +y de : 2 et dx Arowar 1- ye" (c -x*) 1= yet (c -x*) %3D

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Q: ;(y+ 1) e*dx = (e* + 1)y dy, what is F = 1.F. J (-e*y - y)dy? %3D

A: Given: y+1exdx=ex+1ydy

Q: d Evaluate In(et – t²)dt. Briefly explain your answer. dx '1

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Q: Solve the DE using homogenous equations: Let: x = vy dx = vdy+ydv xy dx - (x2+3y2) dy = 0

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Q: where t is a non-zero parameter. Show that dx 28 Given that x=z and y==. 1+r Hence find its value at…

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Q: (e) dy if y = (t +1)(t + 9) %3D dt It%3D3

A: Given that y=(t2+1)(t+9)

Q: dy cos(x +y), when x = 0,y == dx

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Q: dy where C is defined by dx + x2 + y? Evaluate x2 + y? C T(t) =,0 <t< 2n

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Q: dy If x = t – sin t and y =1- cost, then find at t = %3D dx a. -2 b. 0 С. 1 d. 2

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Q: F'ind the differential dy. (a) y = xv8 – x %3D dy =

A: we have to find

Q: Find - given that dt dx x? + y? = 2x + 4y, x = 3 y = 1, and dt 11 dy i dt II

A: Given problem is :

Q: Given a = 2 cost+35 and y =7sin t %3D dx dt dy dt |3|

A: Topic : differentiation

Q: (ztan 2 +1) dx – x dy = 0

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Q: y = secx prove that d² y dx +y 2y

A: For y=sec x, we need to prove that d2ydx2+y=2y3. Here,            y''=secx''      =secxtanx'…

Q: Find the differential dy. (a) y = xv8- x %3D dy %3D (b) y = (1+ x)7 dy

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Q: F(x, y) = (e5x sin(y))i + (ex cos(y))j

A: The given vector field is not conservative

Q: f(x, y) = 2x- xy + y, find (a.)and (b.) at (Xo, yo) didirectly from the definition. dx dy

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Q: f x = sec(z) and y = tan(z) , then dy at z = dx 4

A: We can use the chain rule of derivative in order to solve the given problem.

Q: i x Jl plaly da :J,I Ji ull rcos(@), y=rsin(@) التكامل e-(5x2+2y) dxdy

A: We have to use integration using changing the rectangular axis.

Q: J(y) = (y/(x) + y(x))² dx, for all y E V.

A: In this question, the extremal is given as J(y)=∫01(y'+y)2dx the class  A={y∈V|y(0)=0,y(1)=1}

Q: Find the general solution of this DE using INSPECTION. dnda + +(x? t y?)dx = 0 2 aycdly

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Q: By algebraic substitution, the integral dx dx is converted to: 4+Vx dy y+4 2 dy y+4 ydy y+4 2ydy y+4

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Q: 30. cos(r + y?) dx dy 32. x²+ y² dx dy 2 y

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Q: . Is sin y x² dx + ydy = 0 a linear?  2. Is y' = y+x / x a homogeneous Equation?

A: We have to check whether the given differential equations are linear and homogeneous or not. A…

Q: dy given that y= Vx² (x – 4) dx la) Find %3D

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Q: y = x*, d'y 1 dy dx y dx, Prove that Y = 0. X

A: First we find out dy/dx then d2y/dx2. And put the values of dy/dx and d2y/dx2 in given condition.

Q: Find fa and fy. f(x, y) = 4e dt %3D

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Q: Find the I.F of y(2x-y+1)dx+x(3x-4y+3)dy=0.

A: Given equation is y2x-y+1dx+x(3x-4y+3)dy=0

Q: f(x, y) = sin(z³ – 7y) %3D

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Q: O1f y= x 4x and x = Vzr?+I, find dy /dt when t r2. %3D

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Q: The antiderivative of [Vx²-25 dx is x2 – 25 – sec-lx+ C. A True B) False

A: ∫x2-25xdx

Q: Find dy/dx if y = J' V1 + ² dt. Explain the main steps in your calculation.

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Q: Admit that for all (x, y), af (х, у) — х dx se (x, y) = 4y ду calculate g' (1) = f(2 cost, sint).

A: We have to prove 4y*f'x - x*f'y = 0 Also calculate g'(t).

Q: y sin (r²) dt

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Q: f(x, y) = 8 sin(2xy) dx? dy? dxdy aº f dydx

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