Consider the following initial-boundary value problem 4Ut Uzz, 00 %3D U (0, t) U(2, t) = 0 %3D U (1,0) - sin(TI)+ 4 sin(271) 2 sin By the help of the following relation which is found by the separation of variables method X"(x) X(x) 4T'(t) T(t) = -X the solution of the given problem can be found as U(r, t) = Ane-t/4 sin(vAr) %3D n=1 Find the value of U(1, 1)?

Q: What are the P(x) and Q(x) of y' =y-xy³e2Xrespectively?

A: Given the differential equation is: Since

Q: i) Find the tangent and normal lines to the curve r*y° +sin(xy)+ln(1+y)+sin(2x) = sin(2) at the…

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Q: dx 1)y /sec(3x) 2) y = Vu + 2u , x= u? - 3u

A: Definition used -            Chain rule of derivative -                             ddx(f(g(x))=…

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Q: 4. Find (f-1)'(1) if f(x)=x' +8x+sin(3.x) а. Find (f-1)'(2) if f(x)= Vx' +x +xr+1 b.

A: To find the derivative of the inverse of a function

Q: 1. Let y = x2 + x +4+ Find y'.

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Q: 2. Let F(x) = x s sin (t2) dt. Find F'(x). (a) sin (x²) dx (b) sin (x²) (c) sin (x²) + C (d) 2x sin…

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Q: Given sin(x?y²) + y³ = x + y , using implicit differentiation for partial derivative show dy that dx…

A: Implicit derivative dydx of sinx2y2+y3=x+y

Q: Let f(x) = Then f'(x) = π The equation of the tangent line to y = f(x) at a = can be written in the…

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Q: let h(y)=(y-3y2)5(y2+2)6  h'(y) in simplified form.

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Q: 1. If f(x, y)=3x' cos y– y sin x– 2.x In y- yln x-4x²y, find ôf ôf d²ƒ _ô²ƒ

A: According to question given thatf(x,y)=3x3 cosy-y2 sinx-2x ln(y)-y ln(x)-4x2y

Q: Writing Systems in Matrix Form. In each of Problems 1 through 8, state whether the given system is…

A: Solution:-

Q: 13. Find ƒ. (1)ƒ'(x)=x² + 3√x (2) f(t)=2cost+sec²t,-/2 <t</2,ƒ(π/3)=4; (3)ƒ*(x) = 6x+sinx.

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Q: 8. Find y'if y = x³ +sin (x)

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Q: 4. Determine all possible functions f(x) that satisfy the given conditions. (a) f'(x) = V¤ – 3 x2…

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Q: 12. Show that F(x) = sinx – x cos x+2 is an antiderivative of f(x) = x sinx.

A: Given, F(x)=sinx-xcosx+2 To show that the above function is an antiderivative of f(x)=xsins.

Q: x2 sin3(x) Find ax if y = In \cos(x*).V1+x,

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Q: A) Find the first and second derivatives of the function g(x) = Vx² + 2x .

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

Q: 3 Parameterize y(x) = +1 using x = t sin x Ans=

A: The answer is: y(t) = (-3/sin(-3/sinx) + 1)) + 1Explanation:To answer this question, you just have…

Q: 6. Find the derivatives of the following: a) g(r) = sin(vi) dt %3D g(=) = /" Ve +4 dt

A: Please refer to the image below

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.