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Use finite differences to solve the boundary-value ordinary
with boundary conditions
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EBK NUMERICAL METHODS FOR ENGINEERS
- The general solution to an inhomogeneous second order differential equation is: пх y = Aemx + Benx + ae* + be?x where m = 2.9, n = 1.5 , a = 2.6 , b = 2.2 . with initial conditions y = 2.4 and dy/dx = 2.2 when x = 0 Find the value of B to three decimal placesarrow_forwardThe temperature on a sheet of metal is known to vary according to the following function: T(z,9) – 4z - 2ry We are interested to find the maximum temperature at the intersection of this sheet with a cylindrical pipe of negligible thickness. The equation of the intersection curve can be approximated as: 2+-4 Find the coordinates for the location of maximum temperature, the Lagrangian multiplier and the value of temperature at the optimum point. Wnite your answer with two decimal places of accuracy. HINT: IF there are more than one critical point, you can use substitution in the objective function to select the maximum. Enter your results here: Optimum value of z Optimum value of y Optimum value of A Optimum value of Tarrow_forwardConsider the following linear equations,arrow_forward
- The mass and stiffness matrix of the system is given by: M = and m2 k, +k, -k, K = -k, k, +k, Take m1 =10 kg, m2-5 kg, k1=k3=100 N/m, k2=84 N/m Use modal analysis and determi ne: a) The normalized stiffness ki b) Its eigenvalues and eigenvectorsarrow_forward(3) For the given boundary value problem, the exact solution is given as = 3x - 7y. (a) Based on the exact solution, find the values on all sides, (b) discretize the domain into 16 elements and 15 evenly spaced nodes. Run poisson.m and check if the finite element approximation and exact solution matches, (c) plot the D values from step (b) using topo.m. y Side 3 Side 1 8.0 (4) The temperature distribution in a flat slab needs to be studied under the conditions shown i the table. The ? in table indicates insulated boundary and Q is the distributed heat source. I all cases assume the upper and lower boundaries are insulated. Assume that the units of length energy, and temperature for the values shown are consistent with a unit value for the coefficier of thermal conductivity. Boundary Temperatures 6 Case A C D. D. 00 LEGION Side 4 z episarrow_forward12. Start with Equation IVb.2.8 and obtain Equation IVb.2.9. For this purpose, first ignore the non-linear term compared with the two dominant terms. Then sub- stitute for the velocity and temperature profiles. To develop the integral, consider a case where the hydrodynamic boundary layer is thicker than the thermal bound- ary layer (thus the integral is zero for y > 8'). Arrange the result in terms of = 8'/8 and ignore *.arrow_forward
- The general solution to an inhomogeneous second order differential equation is: y = Aem + Ben + ax? + bæ + c 2. where m = 2.1 n = 1.3 a = 2.3 b = 1.3 c= 2.3 with initial conditions y = 2.6 and dy/dx = 2.2 when x = 0 %3D Find the value of A. Give your answer to three decimal places.arrow_forward1. The general form of linear second-order differential equation can be written in the form: و بار / كلية الهندسة Q4)/ grap dy q(x)y = r(x) d'y +p(x) dx dy b. dx - F(x)y = F(x) x2 dy dx - xy = C. d. r2 d?y dx2 -f(x)y = F(x) 2431)(5-1) 3 (3-21)2 a. (221 -91i) / 169 b. (21 + 52i)/ 13 c. (-90+220i)/169 d. (-7+17i)/ 13 2. Simplify: الحدار المك المراغة 3. If the roots of second order differential equation is complex conjugate, then the gene contain: a. sinusoidal functions and exponentials b. constant and two exponentials c. two constants and two exponentials d. two constants and one exponential 5 4. The order and degree of the differential: 3(3 - + 4y = sinx* are: d²y a. First-order, First-degree- b. First-order, second-degree Second -order, First -degree d. Second -order, second-degree dx2 lo - 2i tisi. 8- 12i 5. The particular solution of (D² + 4)y = cos 2x is equal to: a. sin 2x b. cos 2x 13+159 C. 4 cos 2x d. 4 sin 2x 5-12 lo Best wishes الامتحانية د. مازن ياسین عبود رئيس القسم بن فاضل…arrow_forwardConsider the following ODE in time (from Homework 6). Integrate in time using 4th order Runge-Kutta method. Compare this solution with the finite difference and analytical solutions from Homework 6. 4 25 u(0)=0 (a) Use At = 0.2 up to a final time t = 1.0. (b) Use At=0.1 up to a final time t = 1.0. 0 (0)=2 (c) Discuss the difference in the two solutions of parts (a) and (b). Why are they so different?arrow_forward
- solve this answer is 96252.972 y < -1Sx + 3000 y < 5x In the xy-plane, if a point with coordinates ( a, b) lies in the solution set of the system of inequalities above, what is the maximum possible value of b ?arrow_forwardVerify if the following functions are Linear or not. Support your conclusion with appropriate reason. a) F(x) = b) f(x) =rcos wtarrow_forwardSolve the following differential equation for axial deformation of a bar of length 12mm using Galerkin Weighted Residual method 2022/09/ d² dx² = -0.75(4- x)² One end of the bar is fixed whereas the displacement is 12 mm at the other end of the bar. You may use Matlab for computations. Use the trial function û(x) = Co + ₂x + ₂x²arrow_forward
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