3 I- 0) = [u- w -3 [-1 4] 6.

Q: 1. Express the followings in the form of x + iy where r, y E R.(choose one.) (a) 금+ 부 (b) ( + ) ( –…

A: Rationalise  all the term and calculate to get the final answer.   Actually as per the barteleby…

Q: Find the general solution of the following ODE's: y' + 3x²y = x²

A: The given differential equation is y'+3x2y=x2 It is of the form of linear differential equation…

Q: If f(x,y) = x In(2y³ – 5) + y e* , find ! of and ôx ây

A:

Q: if x~unif. (0,2),find the U.b. of p(lx - mul>= 2)

A: Given X~Unif(0,2) Result: if X~unif(a,b) then  Mean=(a+b)/2 Variance=(b-a)2/12

Q: Find the value of ds / du at u = 2 if s = t2 + 5t andt = (u2 + 2u)1/3.

A: We first find ds/dt and dt/du

Q: If C is the z-axis from the origin to the point (0,0, 2), then So(3j – 3k) - d - dr |

A: If C is the z‐axis from the origin to the point 0, 0, 2, then ∫C 3j→-3k→.dr→=?

Q: Find all values of n such that y = xn satisfies…

A:

Q: . For j = 1,..., N, let Ij = [aj, b;] with a; < bj, and let I:= I1 x -… ·x In. Determine a1. (Hint:…

A:

Q: 8z Find at (2, -2, 1) if By z' +y +z2 = 6xz +6yz +21 eej49 1.

A: Given x2+y2+z2=6xz+6yz+21

Q: 4x obtain the section of the equation Ex xl with the condition u oon X4

A: Let us consider the given PDE is P(x,y,u)ux+Q(x,y,u)uy=R(x,y,u) The equation is solved as Determine…

Q: For x,y E [0,1], prove that xy(1-x-y)≤1/27

A:

Q: Show that the following relationships hold. a) (1+i)™sn[ = Sn+m[ ¯ Sm| b) än+1] = 1+ an] (m) (m), c)…

A: Considering the LHS (Left hand Side) : LHS=(1+i)mSn|¯ RHS=Sn+m|¯-Sm|¯ ( Right Hand Side)…

Q: 1. If 4=x'zi-2y'?'j+xy'zk , find div4 at points (1, -1,1).

A:

Q: 5. By making a suitable substitution, find the general solution y= y(r) of the ODE dy = (y – 4r –…

A:

Q: for z = In(2® + y*). Find

A:

Q: 3. Let f(x, y) = x³ 3x² + xy +7 and u = 0.6i + 0.8j, then find Duf(1,-2).

A:

Q: II. Let U = R, 1 = { x eR | x5 – 26x3 + 25x = 0 }, J = { x € R | |4x + 2| > 10 } and K = { x €R | x?…

A:

Q: Use the fact that 9(z) dz 65 to compute each of the following: g(y) dy = %3! | 9(z) dz 9(z) dz %3|…

A: We use definite integral properties

Q: Find f, and f and evaluate each at the given point. 2xy x, - V+ ay (1, 1) K, ) - Sx, ) - (1, 1) - 29…

A: The solution of the problem is given in the next steps.

Q: Q4/is that, the Sulutton to the differentiel equation dy-tX+4. is it's

A:

Q: Find the value of X and Y in given that; 1 - 2y 4 7 2 2 - 2x 4 - 5 I3D 3. 2.

A:

Q: Find an equatlon Of the ine Through G1,2); Perpendicular to me 2x+ 5yt 8=0 line

A:

Q: Find the value of m that makes: Show that the equation  xex-1=0 has a unique solution on [0,1].

A:

Q: What are the conditions on A, B and C for an ODE of the form Ay'' + By' + Cy = 0 to have solutions…

A:

Q: Obtain (i) P (X 1)

A:

Q: Find the domain of R. R(t) In(1 t) √t+1, e-¹/|t| {(VEFT, 0-10m, bu(1-2)}) t = (1,0,-1) " , if t #0…

A:

Q: 11T 60. Given that sec 12 13T V2 - V6, determine the value of sec 12

A:

Q: If h(z) = 2 – 4, find h(z). %3D

A:

Q: 2ху 3. y' = -x?

A:

Q: Show that the following DE are Bernouli type and solve: 1. y’ – x = y 2. s’+ 2s/t = s 3 t 4

A:

Q: Integrate the following: ,S (x4 – 5æ³ + 6) dæ - 1. So(1- r)*dz su(1– 2.

A: These questions are related to indefinite integration, we will use use integration formulas and…

Q: The ODE z dr - 3y = 0 is a Cauchy-Euler equation and has one solution of the form y =r" for some…

A: The given statement is true.

Q: dt Determine dw if t = (6r* +3r*) -(*+2r-5)'. -s)'. and w =

A:

Q: Verify that u' (t – a) = 8(t – a) by solving the problem x' = 8(t – a); x (0) = 0 to obtain x (1) =…

A: What is Laplace Transformation: Laplace transformation is crucial for control system design. Laplace…

Q: B/ If Ø = (x³y²z) and A = 2xzi - 3y2j+ 3xy²k Find a(A) at ду дх the point (2, -1, 3)?

A: When partial derivative is done with respect a specific variable, then other variables are treated…

Q: (c) Show that |(−1,1)| = |R|, where (−1, 1) = {x € R : −1 < x < 1}.

A:

Q: H.W. a PDE from the followi'ng- @u=f() Form メツ 0u=fcx?ー)

A:

Q: 2s f( 8ts“, ), (x, 9) = 7xy, and af af ;(x, y) : Assume that w= Find and ds %3|

A:

Q: Find e^z in the for u + iv if z = 11πi/2. (use 3 decimals)

A: Given z=i11π2 We have to find ez in the form u+iv

Q: Find the value of k so that the value of S,"(2y – 1) dy is 10. 1 2 3 4

A:

Q: Determine C, and C, so that y(x) = C, e²* + C,e* + 2sinx will satisfy the conditions y(0) = and…

A:

Q: 2. Find the following. 5 – xy? Zy if z = 3x + Y

A: For a function in two variables, the partial derivative with respect to a variable say x can be…

Q: Write a in the form of a = a„T+a„N at t = 0 without finding T and N first if r(1)=(2+3¢+ 3¢°)i+(4+…

A:

Q: Find f, and f and evaluate each at the given point. 2xy KX, y) = + 2y (1, 1) ,(X, y) = F,(1, 1) =…

A:

Q: 6., Given that= 1' + sy = ) and that y = 1 dr (a) show that for values of r close to 0, 1 when = 0,…

A: Let y(x) is a function. Then the expansion of Taylor series of the function y(x) at the point x=a…

Q: 3 (a) Given that T= 2n12g , express I in terms of T, g and n. %3D

A:

Q: Find f, and f and evaluate each at the given point. F(x, y) = e*y?, (In 4, 2) f(x, y) f,(x, y) =…

A:

Q: 3. If V =(3r°y-z,x²+y",-2r:'). Find grad div, at the points (2,-1,0).

A:

Q: Solve the following linear independent and Wronskian. 2. Are the functions y, = e* and y2 = x…

A: Here we have to determine whether the given functions are linearly independent.  We have,  y1=ex…

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.