53. (a) ct = 3cr? (b) (5z + 4cz?) = 5 + 8cz d - ct³ dt d (c) (9c²y³ – 24c) = 27c² y² dy

Q: 7. Find d²y dx² from y = √√x- =√x-²/2

A:

Q: - 9y = 0 y"+5y"+3y' (D2 -25)2 (D2- 4D+ 13)3 y = 0

A: Using method of linear differential equations. As per our guidelines we are supposed to answer?️…

Q: (x*y - 2xy) dy dx 2.

A:

Q: (x2 + y2 + x)dx + xydy = 0 %D

A:

Q: 1. . x + 4 dx x² + 5x – 6

A:

Q: 17 (xr² + 3xy) dx - dy = 0

A: We can solve it by seperation method

Q: 2. Find P.S. of when y (2x²-xy +y ²) dx − x²(2x-1) dy o y = 1/2. X = /

A:

Q: (a) /4 1+3 cos² 1 d.x; COST dx3B

A:

Q: (2x – 3y)dx – (-6y + 4x + 1 )dy = 0

A: Given that 2x-3ydx+4x-6y+1dy=0

Q: (d) fbx ²²+1 dr br

A: We can evaluate the given integral by using the substitution u = ax2 + 1

Q: 2. f(x,y)= 2x + 6xy +y

A: Partial derivative

Q: dy (x³ + 1)(x² – 2x-1+ 1) of y = dx x² + 3

A: y=(x3+1)(x2-2x-1+1)x2+3

Q: 1 -dxdy =? (4+x² + y²)°

A: Here we will use change of variable in double integration

Q: Solve for k such that 1 | k(x² – 2x + 1)dx = 5

A: Given problem:- ∫ k(x2 -2x+1) dx =5  Solve for k

Q: 2 (х + у)dx dy 1. 3.

A:

Q: dy d. x = y + y³cosx dx

A:

Q: (y - x)(y + x)(y? + x2)(xdy- ydx) = x°dx

A:

Q: Find :a. b. 20 fő -1 20 dy (1+y) ² (20-6x) e 20x - 3x² dx

A:

Q: 3. (2x2 + 2)dy + xydx = 0

A: (2x2 + 2)dy + xydx = 0

Q: ¸¸ S'√ 16–16 x² d x = · (a) 16 π (b) π square unit. ©2π (d) 4π

A:

Q: 4. 5 сos(x-3) dx

A:

Q: x/1-y2 B, dx %3D 1+x²

A: dydx=x1-y21+x2

Q: 1 14) Find xe™³x² dx . | 0

A:

Q: 4. Calculate | (6x? + 2ry + 3y?) dx dy.

A: Proceed as shown...

Q: Q2. Solve the following:- r2 1) х4 — 1 2) cot(In x) 4) | (V1- (Inx)² -dp- 3) dx 1- x3 x In x

A: Basic substitution based problem

Q: 2) Find dy of: 7x 5 上 (3-2x

A: We know that  ddx(xn)=nxn-1

Q: •3 dx dy (x +4y)3 36.

A: The given integral is ∫01∫231x+4y3dxdy. The order of integration is dxdy. So, first we have to…