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Solutions to Differential Equations and Numerical Methods

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Added on  2021-04-24

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This article provides solutions to differential equations and numerical methods for engineering mathematics. It covers topics such as separation of variables, characteristic equations, and Euler's method. Additionally, it discusses how experiments can be improved further.

Solutions to Differential Equations and Numerical Methods

   Added on 2021-04-24

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SOLUTIONS TO THE QUESTIONS Task 1) Check excel file Substituting in the given d.edT/dt= -k(90-18)dT/dt= -72KThe first T1= T0+dT/dt= To-72kHence about 10 iterations are performed with h being 0.1 00.20.40.60.811.21.41.61.82050100150200250300350400TTFigure 1: When K= 1Task 2) Method of separation Given dT/dt= -k(T-A)...(i)This has to be first written in this form: A(x)dx + B(y) dy= 0 and then it can be integrated all over.Let dt represented by dy and dt represented by dx. From equation (i) I undertake the following gymnastics: dT/dt= -kT+kAor dT/dt+kT-kA= 0
Solutions to Differential Equations and Numerical Methods_1
implying : dT= -k(T-A) dt in terms of y and x: dy= -k(y-A)dxdy/k= -k/k(y-A)dx , (y-A)dx +1/kdy=0or kdx= -dy/(y-A)integrating both sides: ∫kdx= -∫dy/(y-A)kx= -ln(y-A)mathematically equivalent to: e-kx= y-Ahence T= e-kt +A, (Assume ambient temperature of 250C)t0.010.030.050.070.090.110.130.150.170.190.30.50.70.91.11.31.51.71.9297.6297.8298298.2298.4298.6298.8299299.2TTFigure 1: Temperature graph given K= 1.0
Solutions to Differential Equations and Numerical Methods_2
00.020.040.060.080.10.120.140.160.180.20.40.60.811.21.41.61.82298298.2298.4298.6298.8299299.2TTFigure 2: Temperature graph given K= 0.500.020.040.060.080.10.120.140.160.180.20.40.60.811.21.41.61.82298.7298.75298.8298.85298.9298.95299299.05TTFigure 3: Temperature graph given K= 0.1As K value is reduced from k= 1.0 to 0.5, the graph loses the nonlinearity and becomes almost a straight line. Rate of cooling can be said to be uniform in figure 3 and non-uniform in figure 1 and partly in 2. This is because there is drastic change in slope for the first graph and in graph 3, the change is linear. Task 3: (a) check the graph on the excel file attached
Solutions to Differential Equations and Numerical Methods_3

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