4. Prove that the equation x + sin(x) = 1 has at least one solution in the interval |3D Va + 3 (0, T).

Q: -) Find the general solution of the DE y(4) + 2y" + y = 3 sin t - 5 cos t Use the method of…

A: We can solve this ordinary differential equation by undetermined coefficient

Q: Find the exact solutions of vZ sin (x -)+ 1 = 0 for 0 sxs 47.

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Q: 3. [(3t +r*)sin (21)dt /

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Q: 2. Find an interval where f,(x) = sin2 x and f2(x) = cos² x are linearly independent. %3D

A: W(π/4) = -sin(π/2) = -1 W(π/6) = -sin(π/3) = - √3/2 Since, Wronskian is not zero for a single value…

Q: Is the ODE (sin(x-y)+1)dx=(sin(x-y)-2y)dy satisfied by the implicit solution x+y2=cos(x-y). Show…

A: An implicit function, however, is an equation that defines a relationship between two or more…

Q: Find the critical numbers of f(0) = 2 cos 0 + sin² 0.

A:

Q: Verify that y = 7 sin 2x + 9 cos 2x is a solution of y" + 4y = 0

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Q: 1 If z = 2x2 – 3y with u = x² sin y and v = 2y cos x, determine expressions %3D az dz for and du

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Q: 2. Show that u(x, t) = cos(x – ct) is a solution of Ut + cuz = (0

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Q: x³y"+xy'+4y = cos (2 In x)

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Q: Q#( 1b Compute: [cos (10x)sin²(10x)dx 1 sin°(10x) – A) 30 1 sin°(10x)+C 50 1 cos°(10x)- 1 -cos…

A: We have to solve the given indefinite integral

Q: The f cos 5x cos x dx is 1 {A} = sin 4x - 12 1 1 {B} sin 4x + - 8 12 cos 6x + C cos 6x + C {C} {D}…

A: Given problem:-

Q: 4. The value of fx sin(x²+3) dx Acos (x²+) + C C. cos (x²+)+C B. -cos (x²+)+C D. -cos (x²+) + C

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Q: 6. Find all the values of z satisfying sinh z = = -j1.

A: What is Hyperbolic Function: The hyperbolic functions in mathematics are comparable to the…

Q: au at a²u 28. Solve Əx² U(x, 0) = 3 sin 2+x, U(0, t) = 0, U(1, t) = 0. = where 0 0 together with…

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Q: 2. Find the particular solution of sin y(x+sin y) dx +2x² cos ydy = 0 :3(-1) = 2

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Q: 1. If f(x, y)=3x' sin y- y² cos x- 2yln x– xh y- 4xy , find- of ôf ô²ƒ ô²ƒ

A: Given: fx,y=3x3siny-y2cosx-2ylnx-xlny-4xy2

Q: Find Domain and Range бяц tue Ⓒsin (x+3) +2y x²+2y 3x + y² 3) 3x + Sin (3x) +2y+Z² Ⓒ x+2y cos2x…

A:

Q: Find the particular solution of (cos x x sin x + y2) dx + 2xy dy = 0 that satisfies the initial…

A: Please see the explanation below

Q: 2- Find the largest possible interval in which the given DE has a unique solution tan(x)y" + xy'-y =…

A: Given equation is, tanxy''+xy'-y=x3 ,x0=-2,x0=5 We can write the given equation as,…

Q: (3+ 7cos (x)) sin? (x) de 28. 29. sin° (9y) sec² (9y) dy / tan" (2) cos" (2) dz 30. [tan(2t) – sin°…

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

Q: Consider the following IVP: y" + y' = -cos x; y(0) = a, y'(0) = b Find the general solution. y(x) =…

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Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8); u =

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Q: 4.4. Show that the function f(x) = Sx² cos 1, if x = 0, ૨૦. if x = 0 is a BV-function over the…

A:

Q: 4. Obtain the particular solution of (D² + 4) y = sin x + sin 2x

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Q: iFind the general solution of the equation sin4x 19. Find the general solution of the equation cos2x…

A:

Q: b) Show that the ode (x + tan-¹ y) dx + x+y dy = 0 is exact, then find the general solution and 1+y²…

A:

Q: 3. Find the critical numbers of the function f(x) = sin2 3x + cos 3x , 0 <x < 2/3:

A:

Q: 4. Verify that the solution of the DE d²x dt² + k²x = 0 is x = sin(kt).

A: To verify that the solution of the DE d2xdt2+k2x=0 is x=sinkt.

Q: 5.. Find the general solution to the DE using the method of Undetermined Coefficients: y" – 4y = te'…

A:

Q: Differentiate y= cos y2x +1 sin 2x+1 A. V2x+1 sin 2x+1 B. V2x+1 sin 2x+1 C. V2x-1 sin 2x-1 D. V2x+1

A: Please refer the attached image for complete solution. THANK YOU.

Q: 6. Find the general solutions to the following non-homogeneous Cauchy-Euler equation using variation…

A:

Q: determine the annihilator of the givenfunction. F(x) = x cos 3x.

A: Consider the given function, F(x) =x*cos 3x. Let f(x) is a function, which is defined as,…

Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8)); U =

A:

Q: 1. . Use the method of characteristics to solve the first order equation Uz – 2uy with u(x, x) =…

A:

Q: 11. Find the general solution of uyy - 4xu, = cos 2y – 2x sin 2y + 4x.

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Q: Obtain the General Solution of: (z + 2)* cos yudz - [sin y In(3z 4 6) – d

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Q: D. 1. If (x + y)² = x² + y² 2. If y = sin³ () cos³ () dy - dx X prove that y' = sin² (x) cos(x) 8…

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Q: 7. Evaluate t' sin(2t“) dt .

A:

Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8)); u = i

A:

Q: he Solve the y)-Sy'(x)+8y(x)-4y(x)-xcos(x) equation y,0)-cos(x)+xsin(x)- 41 19 sin(x)+ cos(x)…

A:

Q: 2. Find the general solution of cos x dy + (y sin x – 1) dx = 0

A:

Q: Solve the problem: u – kuxx = e-t sin x , 0 0; satisfying the conditions u(0,t) = 2n, u(t, t) = 4n,…

A: Let the transformation ux,t=vx,t+Ax+B ...............(1)uxx=vxxut=vtthe problem ut-Kuxx=e-t sinx…

Q: Solve the he equation F)-5y'x)+8y(x)-4y(x)-xcos(x) 41 19 41 19 sin(x)+ cos(x) -…

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Q: 3. Show that: • T/2 sin" x dx • T/2 cos" x dx

A: Recall the property of definite integral:

Q: If y = (x – a)/2ax – x² + a² sin¬'(÷ prove that y' = 2/2ax – x² a is constant.

A:

Q: Given that f(x) = V3 cos 5x+sin5x. (a) Express f(x) in the form Rcos(5x-a) where R>0 and 0° < a…

A:

Q: 1. Find Duf at P: f(x,y) = sin(8x – 5y); P (5,8)); u = (i-)

A:

Q: 4 sin (In t) cos`(1nt) dt 5.

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.