(4)Consider again the circuit from HW10# 3. It is written again below for your convenience: 0 I d V dt 쁜은' (a) Show that the eigenvalues are real and equal if L 4R2C (You may use work from the previous problem

Q: Given the boundary value problem y" + Xy = 0, y(0) = 0, y(r) = 0, what is the eigenfunction…

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Q: Let æ(t) = be a solution to the system of differential equations: x2(t) a: (t) a½(t) + 3x2(t) + 2…

A: According to the given information, Let x(t) be the solution of the system of differential equation.

Q: dt dy dt ind the general solution for x(t) and y(t) olve the initial value problem x(0) = 1, y(0) =…

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Q: ) Use the method of eigenvalues to determine the general solution of the following syste…

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Q: For y" + Ay = 0; y'(-n) = 0, y'(r) = 0, answer the following questions. (a) Find the eignfunction yo…

A: we have, y''+λy=0 y'-π=0,y'π=0  a For λ=0 y''=0⇒yx=Ax+B⇒y'x=A ....i Given, y'-π=0⇒A=0 Again…

Q: (2) Use the Laplace transform to solve the initial-value problem: / – 3y = 1+2U(t – 4), y(0) = 0.

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Q: Use the Laplace Transform to solve the given initial-value problem.   2y' + y = 0;  y(0)= -3

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Q: -t yい-yle1-224) yloloo, y'lo).3

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Q: Q.8 (a) Find the eigenvalues and eigenfunctions of the integral equation g(x)=2(2xt-4x²)g(1) dt

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Q: The objective is to identify the solution of the initial value problem using Laplace Transform y" –…

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Q: a?u ди t = at in 00 ax2 BC's: u = 0 at x = 0 for 1>0 u = 0 at x = 1 for t> 0 I.C.: u = 0 at 1 = 0 in…

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Q: Suppose that for a 2 x 2 matrix A, AF= 35 for 7 = • How is the pair (3, 7) called for a matrix A?…

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Q: 1. Reduce to two ordinary differential equations, one an eigenvalue problem, the other having one…

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Q: Solve the initial value problem y′′ − 5y′ + 6y = 6t; y(0) = n, y′(0) = −2 using Laplace transform…

A: According to the given information, it is required to solve the initial value problem using laplace…

Q: Find the eigenvalues and the corresponding eigenfunctions of the following boundary value problems.…

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Q: Q1) The eigenfunctions for an infinite square well of the form (0, 0 <x < a loo, otherwisw V(x) =…

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Q: Solve the eigenvalue problem Lf (x) = Af(x), where k df(x) i dx In the region -a < x< a, subject to…

A: Use the given condition and formula to start. Now apply the boundary condition to determine the…

Q: Q.5 Consider the Sturm-Liouville problem d x=(xy -(xy')+(2-1)y=0, y'(1) = 0, y'(e²) = 0 a) Find the…

A: Given: xddxxy'+λ-1y=0, y'1=0, y'e2=0 We have to find the eigenvalues and eigenfunctions of the given…

Q: 3. (a) Find the eigenvalues and eigenfunctions of the boundary-value problem x2y'' + xy' + (lamda)y…

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Q: -) Find the eigenvalues and the normalized.cigenfunctions of the following boundary value problem:…

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

Q: 1. Show that the eigenvalues and eigenfunctions of the Sturm-Liouville problem y" + Xy = 0, y(0) +…

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Q: Determine the eigenvalues (2„) and eigenfunctions (9„(x)) for the differential equation p" (x) –…

A: Consider the equation

Q: O Find the eigenvalues and eigenfunctions of the boundary value problem y" + Ay = 0, y'(0) = 0, y…

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Q: Consider the boundary value problem y" + Ay = 0, y(0) = 0, y()=0. Assuming X> 0, determine the…

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Q: Find the eigen value and eigen function of the following homogenous integral equation a(t) = ) st +…

A: The homogeneous Fredholm equation is  fx=λ∫abKx,tftdt.  Here, Kx,t is the kernel. The non-zero value…

Q: (2) Use the Laplace transform to solve the initial-value problem: y - 3y = 1+2U(t – 4), y(0) = 0.

A: Given differential equation is y' - 3y = 1 + 2u(t-4)

Q: Q.5 Consider the Sturm-Liouville problem d x- dx -(xy')+(2 – 1) y = 0, y'(1) = 0, y'(e²) =0 a) Find…

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Q: for the following differential equation (assume zero initial conditions), write the state variables…

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Q: Use the Laplace transform to solve the initial-value problem y" - 3y' + 2y = e-4t, y(0) = 1, y'(0) =…

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Q: Q.6 Use eigenfunction expantion method to solve: d cos2 x (xy')+y lIn x, dx y'(1) = e, y'(e) = e

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Q: 5. Consider the ordinary differential equation Prx + Aø = 0 (a) Determine the eigenvalues An and…

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Q: nd the eigenvalues and eigenfunctions of the boundary value problem y" + / + Ay = 0, y(0) = 0, y(2)…

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Q: Suppose that the differential equation x' = Ax is such that A is a 2 × 2 matrix with (1) eigenvalues…

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Q: da 3.2x + 0.75y, dt dy 1.66667x + 1.2y. dt For this system, the smaller eigenvalue is and the larger…

A: step:-1 Given system of differential equation is dxdt=3.2 x + 0.75 ydydt=1.66667 x + 1.2 y Take…

Q: dx 5 dt dy 3 dt

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Q: For a simple harmonic oscillator potential, Vo(x) =-kx ² =¬mo*x', %3D the ground state energy…

A: Given, simple harmonic oscillator potential is, V0→x=12kx2=12mω2x2 For small perturbation H' first…

Q: Consider the eigenvalue problem y" + 2y' + ày = 0; y(0) = y(1) = 0. Show that the nth positive…

A: The auxiliary equation for the given differential equation is:

Q: 1. Use the Laplace transform to solve the given initial-value problem. y' - 3y = 8(t - 5), y(0) = 0…

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Q: Obtain the eigenvalues and eigenfunctions of the following homogeneous boundary value prot + a*xy =…

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

Q: Solve the eigenvalue problem y" + 1y = 0 with the boundary conditions y(0) = 0; y'(1) = 0.

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Q: Consider this problem: 2²y" + ry/ + = 0, y(1) = 0 y(2) = 0. a) Place the differential equation in…

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Q: The eigenfunctions for y" + 2y' + dy = y(0) = 1, y(1) = 0 are {e* sin(nræ)_1O {e2r sin(프) 1(…

A: To obtain the eigenfunctions, find the general solution of the differential equation. If m1 and m2…

Q: Find the eigenfunction assuming the second case for the given boundary value problem. y" +ly = 0…

A: The given problem is to solve the given boundary value problem y"+λy=0, with the given boundary…

Q: る=1 For the system of differential equations below the eigenvalues are: 2=4 dx = 2x+2y dt %3D dy…

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Q: 4x + y dy x + 2y dt a) Find the general solution for x(t) and y(t) b) Solve the initial value…

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Q: Example 7. Show that the integral equation g(x) = 1 (sin x sin 2t) g(t) dt has no eigenvalues.

A: We have to show that the integral equation gx=λ∫0πsinxsin2tgtdt has no eigenvalues. We can write…

Q: Consider the differential equations dx dt 4x+y+z dy dt = 2x + 5y + 4z dz = -x-y dr Show that if it…

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Q: a) Show that the fourth-order Runge-Kutta method, k₁ = hf (ti, wi), h k₁ k₂= hfti+ Wi + 1/12) 2' h…

A: Given:  k1=hfti, wik2=hfti+h2, wi+k12k3=hfti+h2, wi+k22k4=hfti+h, wi+k3 and wi+1=wi+16k1+2k2+2k3+k4…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.