W Find the general solution of the differental cquation nme of the given answas 619+218 = cly-21" 2 Find Hhe solution of the intial alae problem g=t2y', yl)= -1 U3-2 U2- Wnone of the given answes O yu) =
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- 5) Consider four points A, B, C, and D whose geometrical locations correspond to the corners of a square with sides of length 1 mm. Calculate the potential differences (in mV) VAB, VBA, VAC, VCA, VAD, VDA, VBC, VCB, VBD, VDB, VCD, VDC between the points in a uniform electric field of 3 V/m parallel to the two sides (and perpendicular to the other two) of the square. (Show your work)4.The improved Euler's method Use the improved Euler's method with step 0.1 to approximate the solution to the intial-value problem y'=x-y^2,y(1)=0 at x=1.1 and x=1.2State the solution formula for the Cauchy problem for the homogeneous wave equation on R3. Prove that solutions corresponding to a localized initial signal (i.e. initial position and derivative are supported in a small ball) have a wave fore front and a wave back front and that these fronts are close to each other. Conclude that music is possible in R3. Say in words what happens if we consider R2 instead.
- Give a clear and detailed solution of the intergral of e^-ct/m 2)a. Use the 2nd-order Runge-Kutta Method to approximate y(t) with h= 0.25 b. Use the 4th-order Runge-Kutta Method to approximate y(t) with h=0.25 c. Plot both sets {yi} obtained in (1) and (2) d. Determine the eventual population level (as t→∞) reached from initial population.Find the solution of the differential equation that satisfies the given boundary condition(s) . y" - 2ky' + k2y = 0, k≠ 0, y(0) = 1,y(1) = 0
- Suppose U solves the heat equation on the real lineUt = 4Uxx, x ∈ Rwith initial valueU(x, 0) = (4, x ≤ 02, x > 0.(i) Use the Fourier-Poisson formula to give an explicit expression for the solutionU.(ii) Describe the qualitative behaviour of U in this case as t → ∞ and plot outthe solution at several instants of time to explain your answer. What is the limitof U as t → ∞?Find the solution of the differential equation that satisfies the given boundary condition(s) . y" - 2y' + y = 0, y(0) = y(1) = 11. Solve by Cramer’s rule 3x + y + 4z = 11 4x – 4y + 6z = 11 6x – 6y = 3 2. Find the volume of tetrahedron given the following (1, 0, 1), (0, 1, 0), (0, 0, 1), and (1, 1, 1).