(x = s² – t² y = s·t³ z = 2x – 5y,

Q: t2 1- y? dy 6. dt dy ap 2ty? + 3t²y², y(1) = –1 5. r(0) = -2 y(0) = -2 7. dt dt

A: To find the solution of the given initial value problems.

Q: dy Find - for y = du dy du = = [19+ √9 +10t2 dt. 0

A: given y=∫0u9+10t2dtTo find: dydu

Q: Find the particular solution of (y + Vx² + y²)dx – xdy; when x = v3, y = 1

A:

Q: t+1 and y = t2 - 2t + 3, then find t-1 dy If x = dx 1 A t- 1 1 B (t – 1)2 1 C (t – 1)² D -(t – 1)3

A:

Q: (t/E² + y² – y)dt + (vVE? + y? – t)dy = -

A:

Q: х у? dy 1+ y6 ytx dt + t + t3 + (1 – z³)2 ( xy z? dz

A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…

Q: dy Let y = (x2 – vx)4". Find using any method. dx -

A:

Q: a) sec²t + dt 2 + V5 b) ys – 2y c) - dy

A:

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: Find the general solution for y(t) where, dy 10t dt y II

A:

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: b- Yt dt + dBt dY; 1-t

A:

Q: Solve the IVP y" + 2y' + y = U (t – 3), y(0) = 0, y'(0) = 1. -

A:

Q: dy + 2(t + 1)y² = 0, y(0) = – }. 4. Solve the IVP dt

A:

Q: Solve the non-homo. D.E. 1.y" + у%3D Зе2x

A: Differential

Q: If y = t“, then dy 4t³. dt

A:

Q: Solve the I.V.P: dy = 6y²x; y(1) = 6y?r; %3D dx Oy 1 372-8 1 Oy = 8-3z2 Oy 1 8+3z? Oy = 1 !! 8-3z

A:

Q: Use the method of reduction of order to find a second solution to t’y'" + 2ty' – 2y = 0, t > 0 Given…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: 4. Suppose that z = x³y, x= t², and y = t³, find . dt = 12t11 dt 11t10 b) = 10t° dt d) = 9t® dt

A: This question is related to differentiation, We will solve it using given information.

Q: 4 J4- y ktx dy S[S( x, y) dA = [] 3 R 0 2- 2 2

A: Consider the given expression, ∬Rfx,ydA=∫04∫2-y24-yx2-x33dxdy This suggests that the region spans…

Q: 2. Determine the solution of y? dt + (2yt + 1) dy = 0given that given that y (-1) = 8

A:

Q: d² y dt² + 3 dy dt +2y(t) = 2e²¹ ; y(0)=0_

A:

Q: Obtain the genral solution. (D³ +5D²+7D+3)y=0 y=eX(C₁-C₂x) + С3e-³x + С₂e-³x Oy=eX(C₁+C₂x)…

A:

Q: d'y dy 4+y =r²-21 +3 dt? dt

A: We will calculate CF and PI separately then we will add them.

Q: dy Solve I +y= 1 given that y = 2 when t=0. dt

A:

Q: - (t/z? + y² – y) at dt + (y/t² – y² – t) dy = 0; when t 3,y = 4

A: Answer is  ty=3(t2+y2)3/2-363

Q: Find the general solution to t²y + (t + 1) y =t', for t > 0.

A:

Q: Q/ Solve the D.E.: d'y dy 2 + 2 - y = 0 dt2 dt

A:

Q: Find the quadratic approximation at (0,0) for f(x,y) = e^(xe^y) .

A: Solution:

Q: dy Find a solution of x dx y2 - y that passes through the indicated points. (a) (0, 1) y = | 0

A:

Q: t²e*dt dz J2r

A: on solving this we get the

Q: dz Find dt(t=1,s=0) when z = 2x3 + 15y3, where x = t2 - s and y =t+ 2s2

A:

Q: Given z = 20t +24 and y = 90t2 + 103 da (i) dt | dy (i)- dt

A:

Q: %3D y"(t) = 4y(t) – e-tU (t – 1), y(0) = y'(0) = 0.

A: Solution

Q: Solve for y(t) where y(t) + 2 / a – u)y[w)du = t.

A:

Q: Solve :C xy't y = ē xy'- yミ 2.2 S - (+ス)y t. 1十ブえ

A:

Q: *+ 2y = te-2t, y(1) = 0 2. Solve the IVP: dt

A:

Q: Find The Pollowrng in tograls:- Excersises 4 ( 2x + 6y x*) dy de (2 (ay + 29) dë dy + +) dy da x+4

A: As per our Guidelines we can solve maximum 3 question as per request , otherwise we solve one…

Q: 3e2º sec 20 dy determine de Given y = V(0-2)

A:

Q: Find the DE whose general solution is y = Ae Bx A. yy" - (y')2 = 0 B. yy" + (y')2 = 0 C. yy"- y' = 0…

A:

Q: dy 2e3t-2y; y(0) =0 dt %3D The general solution is

A:

Q: If y(t) is the solution of - 2y = 4, y(0) =1, then y(t)= %3D %3D

A:

Q: Show that y=x² is asolution to: xy- 3xy'+4y=0, and then use reduction of order to find a second…

A: Linearly independent solutions of the differential equation

Q: dx dy Explain your solution to the problem: = x + 3y – 2t², 3x +y+t.+ 5 dt ||

A: Given the system of differential equations:              dxdt=x+3y-2t2                  ......…

Q: (y – 2)dr - (x- y – 1)dy = 0 (coefficients linear in two variables)

A:

Q: Solve the non-homo . D.E. 1.y" + у %3D Зе2х

A:

Q: Solve for the DE of: y = C₁+C₂e7x O y'-+7y=0 Oy" - 7y¹ = 0 Oy'-7y=0 Oy" + 7y¹ = 0

A: The given equation is;  y=C1+C2e7x.

Q: C. dt² +y" = 0 dy d. xty" + ху" 3 е* e. t2 dt2. t2 ( - t = 1- sint е.

A:

Q: Q5.a) Find if sec?y = x² + S t² dt Jsiny

A: P.S.: please ask one question at a time.

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.