Derive the D'Alembert's solutiojn to the following Cauchy problem Un = Uxx, - 0 < x < 0,t > 0 u(x,0) = e, u(x,0) = sin²(x),-∞ < x < 0

Q: Obtain the differential equation of the family of lines passing through the center of the conics…

A: First find out the centre of given conic then write the equation of family of lines passing through…

Q: Find the differential operator that annihilates x cos3x + x e³x sin2x.

A: Given equation is  xcos(3x) +xe3xsin(2x) we have to find the differential operator that annihilates…

Q: find the general solution for  2 Z_xy+ Z_yy =cos( y) +1

A: Firstly find a homogenous solution and then find a particular integral . hence complete solution =…

Q: Express a general solution to 2x(1 – x)y" + (3 – 10x)y' – 6y = 0 Using Gaussian hypergeometric…

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Q: This is the solution of the Cauchy - Euler equation xy" + 5xy' + 20y = 0 %3D O y = C1x4 cos(4 ln(x))…

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Q: 1. Consider the equation u +Uy = 1, with the initial condition u(x, 0) = sin(x). (a) What are the…

A: From the given information. The given equation is as follows. The initial condition is as follows.

Q: Solve using Laplace Transfrom Method: y'''+3y''+3y'=y       y(0)=7,   y'(0)=-7,    y''(0)=11

A: Given function is, y'''+3y''+3y'=y    y(0)=7,   y'(0)=-7,    y''(0)=11 Now, using the…

Q: 14. If f(z) = 2 + ² -r', then the equation S(z) = 0has a solution on the interval A [1,2] B (0,1] c…

A: Given equation, fx=2+x2-x3, To find the interval on which the equation fx=0 has a solution.

Q: 2yy" = 1 + (y')² 2

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Q: (D³ + 4D² + 9D + 10)y = -24e¯* when x = 0, y = 0,y' = -4, y" = -10 ANSWER: y =e-2* + e-cos (2x) -…

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Q: 2. Solve the following Cauchy problem: u = u -0 0, u(x, 0) = 1 -1, u(1,0) = 4r, -00 <I< 00.

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Q: 4Ux+Uy=U² , U(x,0)=1/1+x²

A: In the given partial differential equation with the given initial condition. 4ux+uy=u2u(x,0)=11+x2

Q: a) b) d'y(t) _2d'y(1) dªy(1) _4dy(t) + 5y(t) = 6x(1) dt' di' dt¹ dt d'y(1) di' dit Property +2dy(t)…

A: Linear: A differential equation is linear if the unknown function and its derivatives are to the…

Q: When does an nth order, homogeneous, Cauchy-Euler equation anty(n) + an-1tn-ly(n-1) + +aity' + aoy =…

A: Given: nth order To find: Cauchy-Euler equation

Q: 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard…

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Q: find dy of X sin 2y =YCO52X

A: Given: x sin 2y=y cos 2x To Find: To find dydx.

Q: 4 a. Consider the i.v.p x' = t^(2) + cos(x), x(0) = 0. Verify that the hypothesis of Cauchy Picard…

A: a. Verifying Conditions of Cauchy-Picard Theorem  b. Estimating the Interval of Existence

Q: 1. Solve the initial value problem on the real line Utt - Uzz = sin x U(x,0) = 0 Ut(x,0): = sin x

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Q: tan - de 3 3 sec

A: According to Bartleby's guidelines we can solve only 1 question at a time. Here's the solution for…

Q: Are these equation separable 1.s2+ds/dt = (s+1)/st 2.dr= b(cosx dr+ rsinxdx) 3.y'=ye(x+y)/(x2+2)

A: The objective of the problem is to determine whether the given equations are separable. A…

Q: 5) S V2 + sin 3t · cos 3t dt

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Q: 32) Find ур(0) for the equetion y (4) - y" - 2y + 2y = 3xe³ + cos x

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Q: 3. Solve the IVP: siny(x + siny)+2x²cosy y'=0; y(-1)= π/2 (Subst. first and then Bernoulli's)

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Q: 5. If xy + 3y2 = 2, evaluate the values of (a) y at the point (-1, 2) and b Find dy for the…

A: Given: (a) To evaluate y' at the point (-1, 2), (b) To find the derivative dy/dx?

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

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Q: Consider the IVP y"(t)-2y'(t) + 5y(t) = 0, y(0)=-1, y'(0)=3 (a) The characteristic equation of the…

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Q: 2. Determine whether sin x or cos x is a solution to y" - 2(cot x)y' + (1+2 cot² x) y = 0 Hence use…

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Q: 1. Find all singular and all ordinary points of the DE (0.5x²+2x+25)y"+xy'+y=0

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Q: Find the general solution of y" + 12ly = 0. Oy(t)=c₁e-12¹+c₂e Oy(t)=c₁e¹¹+c₂e Oy(t) = c cos 11t+c…

A: This question is about ordinary differential equation.

Q: Rocky Mountain Chocolate Factory (RMCF) founder and president Frank Crail employs 220 people in 361…

A: for net price  equivalent rate determine first find the complement of the chain discount rate…

Q: 1. у' + ху % 3 y(0) = 0

A: According to question given thaty'+xy=3y(0)=0

Q: Which of the following is a fundamental set of solution of y" – 2y' + 5y = 0.

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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.