a2u 1. Solve the equation ax2 1 0'u for 00, and 4 at2 subject to the boundary conditions L a) u(0, t) = u(2, t) = 0, b) u(x, 0) = 6 sin Tx - 3 sin 4Ix du c) (x, 0) = 0 at

Q: 2. Solve the boundary-value problem = 4, y(0,t) = y(5,t) = 0, y(x,0) = 0, y.(x,0) = 5 sin rx. %3D

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Q: 2 Solve S dy Use u substibation with u=y² ¢ CLearLy show gamma function r (n+2) = nTn) propeaties

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Q: 4. Solve the forced undamped mass-spring problem z" + 9x = sin(3t), x(0) = 2, x' (0) = 0. Graph the…

A: in the given problem we have to solve the give forced undamped mass spring problem…

Q: 3. If only one person is on the swing in the image below, it is comparable to a point load of F =…

A: Given; F=500N x=0.5m length of the branch= 1.5m I=1* 10^-6 m^4 E=2*10^9 Nm^-2   the boundary…

Q: A rectangular steel sheet is bounded by the axis x = 0, y = 0,x = a and y = b. The temperature along…

A: Given differential equation denoting the temperature of the rectangular steel sheet…

Q: 4. (6 points) Using separation of variables, solve the differ- ential equation, (1+7)dy dx y Use C…

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Q: A flexible string of length t is tightly stretched between x = 0,x = n, on x-axis, its ends being…

A: To solve the problem we use the method of Laplace transformation also Laplace inverse transformation…

Q: Solve the initial-boundary value problem using separation of variables method for the Following…

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Q: 6. Solve the boundary value problem, or show that it has no solution: П y" – 4y = sin(x), y (0) = 0;…

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Q: a1/ Find c; and ey so that ytx) = c; sin x +cz cos x will satisfy the given conditions. Determine…

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

Q: You get to a step that you found a solution of the form: u(r,0) = > An(rn – r=")sin(n0), n=1 where…

A: In this question, we a general solution of Laplace equation is polar form is…

Q: 6. Let u (x, t) be the solution of the initial-boundary value problem U = 0 0, Uz (0, t) = ux (4,…

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Q: Q6- For following partial differential equation, answer only a or b: a- Write down the explicit…

A: Solution: (b) Transform into polar coordinates The equation is δhδt=δ2hδx2+δ2hδy2…

Q: Consider a spring-mass system with an external force 38(t – 4) + sin(t), which is described by y" +…

A: Hello. Since your question has multiple sub-parts, we will solve first three sub-parts for you. If…

Q: The solution for the differential equation (D2 + X²)y = 0 with boundary conditions y(0) = 0, y(3) =…

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Q: A storage tank contains a liquid at depth y where y = 0 when the tank is half full. Liquid is…

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Q: Show that xo = 0 is a regular singular point of the equation xy" – 5 - +x ) y' + 3y = 0). Determine…

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Q: 12 Solve the boundary-value problem du 0 0 at + sin x %3D u(0, ) = 400, u(T, t) = 200, r>0 u(x, 0) =…

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Q: You can verify that the differential equation: – 4x²y" + x(x + 8)y – (x + 8)y = 0, x > 0 has…

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Q: 1. y" - 2y' - 3y = 5e¹, y(0) = 1, y'(0) = 0 2. y" + 2y + 5y = 3 sin(2x) 3. y" - 4y + 4y = -7x+4x²,…

A: Since you have posted multiple questions. As per guidelines we"ll solve only  first question. If you…

Q: a) A flexible string of length is tightly stretched between x = 0, x = π, on x-axis, its ends being…

A: The given problem is to solve the given partial differential equation with given initial conditions…

Q: 4) Solve via separation of variables: u(0, y) =0 ди -(2, y) = 3sin a'u a'u +5 sin =0 ; 6. S0<x<2…

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Q: Show that an implicit solution of 2x sin?(y) dx – (x² + 17) cos(y) dy = 0 is given by In(x2 + 17) +…

A: To find- Show that an implicit solution of 2x sin2y dx - x2 + 17 cosy dy = 0 is given by lnx2 + 17 +…

Q: A rectangular steel sheet is bounded by the axis x = 0, y = 0, x = a and y = b. The temperature…

A: Answer is given below

Q: dY/dt =  ( 2  2 )Y.             with initial condition Yo=(1,1)              ( -4  6 ) a) find…

A: Given dydt=22-46y  with initial condition y0=1,1

Q: 24. Radiation. Find steady-state temperatures in the rectangle in Fig. 296 with the upper and left…

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Q: 22. Consider the boundary-value problem a²u əx² a²u + = 0, 0, 0<x< 1, 0 <y< ay² u(0, y) = un cosy,…

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Q: 18- Given with boundary condition y(0) -1.5, y()-0, using 6 equidistant nodes +Sy- cos(3x) - 0 use…

A: The complete solutions are given below

Q: It can be shown that y, = 2 and y, = cos²(9x) + sin²(9x) are solutions to the differential equation…

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Q: 2. Solve Un = Uxx X > 0, t> 0 with initial and boundary conditions given as u( x,0) = 1– exp(-x),…

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Q: Match the differential equation with its direction field. y'- sin(10x) sin(10y) 20) -0.3 1-0.2 1…

A: Remark: We know that the slope field is a graphical representation of a differential equation. This…

Q: Determine the largest interval (a, b) for which Theorem 1 from section 6.1 guarantees a u solution…

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Q: ди 10²u = It 40x²¹ together with the boundary conditions u(0, t) = 0 and u(, t) = 0 for t≥0 and the…

A: Given eq. is ∂u∂t=14∂2u∂x2      (1)  u0,t=0, uπ2,t=0  for t≥0 ux,0=xπ2-x  for 0&lt;x&lt;π2

Q: 2. Consider a homogeneous differential equation, p(т, у) dx + q(z, у) dy %3D 0, where the…

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Q: The dynamics of the variable ywith respect to variable xis described by the following stiff ODE: dy…

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Q: Use Heun's method with h = 0.25 to solve the initial value problem, on the interval from x = 0 to 2:…

A: Given dydx=fx,y=yx2−1.1y Heun’s method use a pair of equations in each step. Predictor:…

Q: Find solution to hon-homo geneous heat problem ut = Uxx + Cos(HX) ; +70 17X70 u(x,o)= f(x); OLX <I…

A: We will use basic knowledge of PDE theory and analysis to answer this question.

Q: 3. A uniform rod of length 10 meters is held at both ends. The temperature u(x, t) at any time t of…

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Q: Solve the partial differential equation: a²u azu by the method of separation of variables for a…

A: Given: ∂2u∂t2=c2∂2u∂x2 To find: ux,t=0

Q: The first condition allows computation of the spring constant, k. It follows that the damped…

A: we will first open the complementary solution associated with homogeneous part of the equation then…

Q: The differential equation dy 4e" cos(y) + 3 dr 4e* sin(y) – 6y In(x) + 3y? Has solutions of form…

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Q: 5.Express this model of an electric circuit d²y dt² dt As a pair of first order equation +6+5y=sin…

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Q: Let u (x, t) be the solution of the initial-boundary value problem Ut 0 0, Uz (0, t) = uz (4, t) =…

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Q: 3. A compression rod of length l is loaded by a sinusoidal force as shown in Figure. The bending…

A: Using the method of undetermined the coefficients for particular solution for nonhomogeneous part.

Q: 4. Solve the heat equation du +et sin(3.x) with the boundary conditions u(0, t) = 0, and u(7, t) = 1…

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Q: Consider the differential equation ry(3) – 6ry' + 12y = 0. Find distinct values of r, say r1, r2,…

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Q: Suppose we have xy – w = y - w - 3z = 0 w³ + z° – 2zW %3D - where x, y, w arc endogenous variables…

A: When the partial differentiation is done with respect to a specific variable, then all other…

Q: Consider the initial value problem * = (tan (z + y))² y (0) = 0 %3D What is limg-1 y (x)?

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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.