Consider the temperature in a uniform rod of length 2 if the initial temperature is f (x) = x %3D and if the end x = 0 is insulated while the other end is maintained zero degree tem- perature. (a) Find an initial-boundary value problem of PDE to model the temperature function u (x, t): (b) Solve this PDE.

Q: Consider dy = 4-t-2y dt y(0) = 1 Using At 0.1 find y(0.1), y(0.2), … ,y(1) using Runge Kutta Method.…

A: Given first order differential equation is dydt=4-t-2y, y(0)=1 So f(t,y)=4-t-2y, y(0)=1 Step size…

Q: A thin bar of length L = 3 meters is situated along the x axis so that one end is at x = 0 and the…

A: Given: Length of the bar L = 3 meters  is situated along the x axis so that one end is at x = 0 and…

Q: Find C1, C2, and Ca so that the given functions will satisfy the prescribed initial conditions. 4a.…

A: We will find out the required values.

Q: A large tank contains 20 liters of pure water. Then a process begins where water containing 50 grams…

A: We have to set up the equation

Q: The function u(x, t) = exp(-x²) is a solution of the heat equation 4rt au au² 30-3422 at ax² Use…

A: Given the solution ux,t=14πtexp-x24t of the heat equation ∂u∂t=∂2u∂x2.

Q: Let Y(t)= X(t) be the solution of , ly(t). 6 - 1 Y' = -2 Y, Y(0) = 8 5 4 Write y(T/10), round your…

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Q: Use Secant method to to approximate the solution of f(r) = z+ cos z =0 accurate to within 10-3,…

A: The secant method is used to calculate the root of the function. The formula for the secant method…

Q: 2. Find the particular solution using the initial condition. ry'-y = x,y(1) = 7.

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Q: d²y =t+ cost di2 Q.4(20 Points): Find a function f(t) such that and such that y(0)%=D1 and y (0)=2.

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Q: Use the Runge-Kutta method to approximate x(0.2) and y(0.2). First use h = 0.2 and then useh = 0.1.…

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Q: Find a function y=P(t)that is a solution of y'=.02y with initial condition y=100 when t=0.

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Q: Solve the IVP Cost y +7Y y(T) = 0, t> 0 %3D t2

A: Given IVP is,  y'+2ty=costt2, yπ=0, t>0.   It is of the form of y'+Pty=Qt. Here, Pt=2t and…

Q: Approximate y(0.45) using Heun's method with the equation y' = 1.2x + 4y by using a stepsize of 0.15…

A: Let us find the value of y at x=0.45 using Heun's method in the next steps.

Q: 2. A 0.2kg weight stretches a spring 0.1m. The system is submerged in oil with damping coefficient…

A: Given the mass of the weight is 0.2 kg and the weight stretches the spring 0.1 m. The system is…

Q: The motion of a mass-spring system with damping is governed by y''(t) + by' (t) + 36y(t) = 0; y(0) =…

A: y''t+by't+36yt=0y0=3y'0=0b=10

Q: The motion of a mass-spring system with damping is governed by y''(t) + by' (t) +9y(t) = 0; y(0) =…

A: Given that the motion of a mass-spring system with damping is governed by y''(t) + by' (t) + 9y(t) =…

Q: Solve the following Initial Value Problem: y' sinx = y(y − 1), y = y(x),y (7) = 2.

A: Here we have to solve the given initial value problem.

Q: (x² + 1) d x + 3x(y - 1) = 0, y(0) = 7 + 1) ( ² 2 y = 1 + 6(x² +1)

A: The given differential equation is; (x2+1)dy/dx +3x(y-1) =0, y(0)=7.

Q: Use Euler's method with specified step size to determine the solution to the given IVP at the…

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Q: Problem 8 Find the solution of the given initial value problem. Sketch the graph of the solution and…

A: Consider the given equation. y''+4y'+3y=0 Write the characteristic equation.…

Q: sove Solve the following initial value y' + 3y=e, y (0) = 7

A: Please see the below picture for detailed solution.

Q: Determine the longest interval in which the given initial value problem is certain to have a unique…

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Q: 4. Solve the IVP y' = 2x-3y +e¯* y'= 2x- 3 y +e * with y(0) = 1 y(0)= 1

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Q: A 50 gallon tank initially contains 20 gallons of pure water. Salt solution, at 0.5 lb/gal, begins…

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Q: Determine the y(t) for the following initial value problem: y" – 2y' + 5y = 8e'sin2t -

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Q: Solve the IVP for y(x): dy 2 + ²y = 12y³, _y(1) = 1 dx x y(x) =

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Q: A horizontal beam of unit length under load w(x) satisfies the system: y" = M, M" = w(x), x = (0,…

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Q: Consider the following initial value problem: = sin(y +t) +e - sin(e' +t), y(0) = 2 Show that there…

A: Given the initial value problem, y'=siny+t+et-sinet+t, y0=2   We will make use of the existence and…

Q: (4) Suppose the acceleration of an object is given by a(t) = t m/s. Its initial velocity is v(0) = 5…

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Q: A 2-kg mass is attached to a spring hanging from the ceiling, thereby causing the spring to stretch…

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Q: Consider IVP: y' = y² + t², y (0) = 1. Using Euler's method with time step h find y (2) . 0.5.

A: Euler's method yn+1=yn+h⋅f(tn,yn)where  tn+1=tn+h