f2 W3 ea f3 f4 U3 e7 f6 es WA Ws e6 G' G 4.

Q: QI/ Evaluate: 1-S(y3" + y2n + y")(2y2" + 3 y" + 6)i dy.

A: Proceed as shown below...

Q: Evaluate (4D5 – 23D3 – 33D2 – 17D – 3)y = 0 - 3x a.y = (C1 + C2 + C3x²)e¯* + (C4x + C5)e7 %3D 3x b.y…

A: Given ODE is: (4D5-23D3-33D2-17D-3)y=0

Q: ey+y°.

A:

Q: ions: a). f(2) = ea %3D 22 +9'

A: As you have asked two unrelated and different, we shall answer first question only as per the…

Q: а) f.(x, y) b) f (x, у) ху 1 3 f (x, у) 16x4y4 Зу'х + 5ху

A: Consider the given function

Q: 2. f(t) = (t + 2)² et

A:

Q: (x, y) = (0,0) %3| = {r3+y3' 0, (b) f (x, y) (x, y) # (0,0)

A:

Q: e3x dx (1-2е3х)2

A: ∫e3x1-2e3x2 dx Put, 1-2e3x=u⇒0-2·3e3x=dudx⇒-6e3x dx=du⇒e3x dx=-du6

Q: 1. Evaluate: e 3z I = dz (e² + e5z)2

A: given integral I=∫0∞e3zez+e5z2dz claim- evaluate the integral

Q: A 1000 16 (*)° + 5x(»*)° + x*y* = 0 B 0007-16 u101210007- =0 0007 y*(y"アーメy=0 210 D 210 210007-16…

A: The solution is given as

Q: Find the natural domain and graph the functions in Exercises 15–20.

A: (15) Consider the provided function, fx=5-2x Find the domain of the function. The domain is the set…

Q: (w³ + wz² – z)dw + (z³ + w²z – w)dz = 0

A: Let us consider a differential equation  Mdx+Ndy=0  To check the equation is exact, determine…

Q: 3) ATUV = AGFE U G F E V ZU = ?

A: Here we use the property of symmetry of triangle.

Q: Evaluate (D4 + 3D³ + 2D²)y = 0 a. y = C1 + C2x+C3e¬* + C4e' b. y = C1 + C2x+C3e¬* C.y = C1x +…

A: To solve the given differential equation.

Q: min z= A x, +B X2 s.t. X+*2 S6, X1 2 4, X1, X2 2 0.

A: NOTE: Refresh your page if you can't see any equations. . subject to  After introducing slack,…

Q: h(r) = - 6/(r − 5)2   concave upward        concave downward

A: When the second derivative is positive, the function is concave upward. When the second derivative…

Q: (D6 – 3D5 + 3D4 – D³)y = 12 A. y = c1 + C2x + e*(c3 + c4x + c5x²) – 12 B. y = c, + c2x +e¬*(c3 + c4x…

A:

Q: (s – s2 – t2)ds + (t + s2 +t2)dt = 0

A:

Q: Describe and sketch the surface given by the function.

A: Concept: The calculus helps in understanding the changes between values that are related by a…

Q: f (x, y, z) ) = x² + y² + z²; x* + y* + z* = 1 2. 4 || ||

A:

Q: 3 - 9, y(0) 33 - 30 – 13 a(t) = y(t) = %3D

A: Given,       x'y'=53-30-13xy     x(0)=-9, y(0)=33

Q: 교고 + z-2 I+ -2, b) f(x) = I+2 -3, I = -2;

A:

Q: Find fx (x,y) and fy(x,y). Then, find fx(-1,-4) and fy (-3,2). 32 f(x,y) = 4x³y² fx(x,y) =

A: We have to find partial derivatives of Given function.

Q: e x + > 2 , x 0

A: The given equation isx+1x>2x≠0

Q: 1² – y (a) f(r,y) = 3 (b) g(x,y) = e"-z

A: Now , We will check the functions .

Q: | f(x) dx = -2, f(x) dx = -7, g(x) dx = 9, g(x) dx = -5, %3D

A:

Q: t < 0; 4. f(t){t, 0 <t < 4; t 2 4. rof (6,

A: As per the company rule, we are supposed to do only one problem. Kindly repost the second problem.

Q: f (t) = 6e 5t + e3t + 5t3 – 9 As5 + Bs* + Cs3 + Ds² + Es + F F(S) Gs6 + Hs5 + Is* + Js³ + Ks2 + Ls +…

A: This is a problem of Laplace Transform.

Q: c) L{tU(t – 2)} d) L{sen t H(t – "/2)} e) L-1se- s²+16, e-2s

A: Given: a) Lt ut-2 b) Lsint Ht-π2 c) L-1se-2ss2+16

Q: (ii) (2 – t4)4 dt *xª – 5x³ + 6x² – 18 dr g3 – 3x² (iii)

A: 2)

Q: f(u) — (4и + 5)(7uз — 2и) 4 b. h(t) = - a. %3D f(2) = -6z3- 2: C.

A: 1) f(u)=(4u+5)(7u3-2u)      =28u4+35u3-8u2-10udduf(u)=28×4u3+35×3u2-8×2u-10        [ddxxn=nxn-1]…

Q: f(x) = e−x,    n = 5

A: Given:          fx=e-x ,n=5 A Maclaurin series is given by fx=∑n=0∞fnan!xn Here fx≈Px=∑n=05fnan!xn…

Q: determine L−1[F].

A:

Q: 3. f(t) = -2t + 3t- 6 6. h(s) = s5 1+1 9. h(1) = + 21 + 2

A: To find the derivatives of the following functions 3) f(t)=-2t2+3t-6 6) h(s)= s4/5 9)…

Q: 2) 2 A) 1 C) i 1) f A) 1 C) i B) -1 D) -i B) -1 D) -i

A:

Q: 4 e 2t8(t + 3)dt 00 | 4e -"6(t + 3)dt =| - 2t "8(t + 3)dt: - 00

A: Definition of delta function  Then spilting the integrals we can get the answer.

Q: 4x1 -5x2 +3x3 +3x4 4 -I1 +x2 +3x3 +3x4 4 3x1 -4x2 +6x3 +6x4 8 -3x1 +3x2 +9x3 +9x4 12 +s +t

A:

Q: (d) fa(x,4) = (e) f,(4, y) = (f) fy(x,4)

A:

Q: (c) y = 1, y = 1/2, x = 2 (d) y = √x, y = 22, x=9 (e) x = 2y², x = 4 + y² (f) x = 1 - y², x = y² – 1…

A: The area between any two curves is given by formula A = ∫ [f(x) - g(x)] dx  with limits from a to b…

Q: C(-1, 3) and D(-5, 7) Vex, - x,} +(y, -y.: d%3= X, = _ ; X, = iY2=_ Y=

A:

Q: Evaluate (4D5 – 23D3 – 33D² – 17D – 3)y = 0 - - | | 3x a. y = (C1 + C2 + C3x²)e¬* + (C4x + C5)e7 %3D…

A:

Q: f (t) = 6e-6t + et + 5t³ – 9 As5 + Bs* + Cs³ + Ds² + Es + F F(S) : Gs6 + Hs5 + Is4 + Js³ + Ks² + Ls…

A:

Q: Find fx (x,y) and f, (x,y). Then, find fx(1, -2) and fy (-3,3). f(x,y) = -2xy +9y² + 10 fx(x,y)=

A: We can find the answer as below.

Q: Evaluate (D4 + 3D³ + 2D²)y = 0 a. y = C1 + C2x+C3e¬x + C4e¬2x b.y = C1 + C2x+C3e¬* C.y = C1x +…

A: We have to solve the problem

Q: 3x d.A, A= {(x; y) E R² : y E [0, 1], x € [y, e®]} .

A: Given: A double integral ∬A3xdA, A=x;y∈R2:y∈0,1, x∈y,ey ∬A3xdA=∫01∫yey3x dxdy=3∫01x22…

Q: 11. T: R³ R³, T (x, y, z) = (x + y, x − y, z)

A:

Q: e 3z -で dz (ez + e5z)2 I = 0,

A:

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.