What is the longest interval for which the Existence and Uniqueness theorem guarantees a unique solution to the initial value problem + (Int) y' + y = y (1) = 1, y (1) = 0? O (0,1) D (0,2) D (1,2) D (0, 0)

Q: u²(x) = -x + √(x-1)' u(t)dt, u (0) = 0,u (0) = 1.

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Q: Q7/ Express in the form x +jy (1) e(r/2-j) (2) In(3 – j) (3) sin(t/4+ j2)

A: (1)  ejπ/2-j=ejπ/2-j2                =ejπ/2+1                                         j2=-1…

Q: 2) Let y = c,x + C,x ln x be the general solution to x'y" – xy' + y = 0 on(-∞,). Find c, and c, such…

A: Given that the below equation is a general solution of x2y''−xy'+y=0. y=c1x+c2xlnx To find the value…

Q: What is the linear approximation of this function at (1,0) F(x, y) = (F(x, y) = (x² cos(x+y)) x²y 0…

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Q: 4. Use the method of variation of parameters to find the general solution of the equation

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Q: A local target store bought 116 pizzas at Pizza Hut for its holiday party. Each guest ate 1/6 of a…

A: Let x be the number of pizzas bought by the store and y be the number of guests target have for the…

Q: u²(x) = -x + = = = √" (x −1 )* u(t)dt, u (0) = 0,u'(0) = 1.

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Q: show that $(t) > $ Ct) for tocteti consider the differentiàl equations dt + P,(t)X = 0 CA) and P(t)…

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Q: Determine the largest interval (a,b) for which Theorem 1 guarantees the existence of a unique…

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Q: 2. Consider the problem u"(x) + u'(x) = f(x) u'(0) = u(0) = [u'(1) + u(1)], with f(x) a given…

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Q: Let A 22 = - (²3) 11 Find the solution of the problem: X Y = AY + [*]. Y = (0) -X

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Q: x?y" – 4xy' + 6y = 0, y(-2) = 8, y'(-2) = 0

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Q: Solve the boundary-value problem y" – 8y' + 16y = 0 y(0) = 2 y(1) = 0 %3D - Use a as your…

A: We First find the auxiliary equation then its roots  at then provide the general solution of the…

Q: find the interval of the inicial value

A: Given Data: Let us consider the given data, t-6y'+t-2y=lnt+t-1 , y3=-4. To find the interval of…

Q: If f(x) = > anx", where the coefficients in the ex- n=0 pansion satisfy En(n + 2)anx" + (n –…

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Q: If y" – 2y' + 3y = 0, y(0) = -2, y'(0) = 1, then L{y(t)} = Y(s) = a) 2s+3 82–2s+3 b) 2s+5 g2–2s+3 c)…

A: The given differential equation is y''-2y'+3y=0 ....................(1) Let y=emt ⇒y'=memtand…

Q: Q3) (a) Consider the boundary value problem: x"(t) = g(t, x(t)), t = [0, 1], x(0) = 0, x(1) = 1.…

A: Given  Boundary Boundary Value problem is

Q: Suppose that y' = f(y) has no constant solutions, where f : R -> R is infinitely many times…

A: Consider the provided question,

Q: Consider y" + 4y' + 5y = ae -3* 8 (t - 31), y (0) = 1, y (0) = -2. Зл Find the value of a such that…

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Q: u²(x) = -x + ² = f (x −1 ) 'u (1)dt, u (0) = 0,.u '(0) = 1.

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Q: Solve PDE 1 = e=*t u. u̟ =e¯*t Xx k (o0) u(0,t) =0 u(L,t) = 0 u(x,0) = L -x?

A: The given PDE is,uxx-1kut=e-xt

Q: not Show that ([0,1]_is in the l' metric d (fig)= (51f6)-96) ³d x) ² fumpluts

A: The given set is X = C[0,1] and the metric defined on X is L2 metric which is defined as  df, g =…

Q: x²dy + (y² + 3xy + 4x²) dx = 0 when x = 2, y = 1

A: To obtain general and particular solution satisfying initial condition indicated byusing homogenous…

Q: If y1 and y2 two solutions of (1 – a²)y" – 2xy' + n(n+1)y = 0; -1<x < 1 Show that W (y1, Y2) : 1-…

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Q: tiven the IVP y" +2y" - y' - 2y = 0, y(0) = 1, y'(0) = 2, y"(0) = 0 use the existence nd uniqueness…

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Q: Find all functions that form a fundamental set of solutions for t2y" – 2ty' + 2y = 0. O y(t) = et O…

A: We have to solve given problem:

Q: Solve the 2 dy JX Y(1)=1 S Following initial value problem x²=2x² x(y +1)+(y+1) X T

A: The initial value problem is x2dydx=x3-2x2xy+1+y+1 with y1=1. The objective is to solve for yx. The…

Q: Solve the separable I.V.P y' (x2 +1) = x (y² + 4), y(0) = 0. y = 2 tan(2 + In(x + 1)) y = 2 tan(In(x…

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Q: What is the general solution of y"'-3 y"+3 y'-y = e* – x-1

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Q: If y is a solution of y(x)-[(x-1t)»(?)dt = 1, then which of the following is true? (a) y is bounded…

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Q: Determine the interval of existence and uniqueness without solving

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Q: The value of c of f(x) = 2-, in the interval (1,3) such that f'(c) = %3D f(3)-f(1) %3D is 3-1 V340…

A: Given fx=2-3x given interval 1,3 f'c=f3-f13-1

Q: Solve the 2 dy JX Following initial value proble x-2x 4 + T XI 4

A: We have to solve the given initial value problem.

Q: solve  utt- 4uxx=x2          -infinity 0  subject to initial conditions: u(x,0) = x  ut(x,0)=0

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Q: 07/ Express in the form x + jy (1) e'(x/2-j) (2) In(3 – j) (3) sin(r/4+ j2) -

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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.