Fundamentals of Mathematical Methods and Mechanics Question 1 5 p...

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Fundamentals of Mathematical Methods and Mechanics Question 1 5 pts A differential equation is given by, #== with the initial condition y(0)=1. Find the value of y when x=1.70. Give your answer to 4 decimal places. Question 2 5 pts A differential equation is given by, = xc with the initial condition y(1)=2. = Find the value of y when x=1.57. Give your answer to 4 decimal places. Question 3 5 pts By the method of substitution, evaluate the definite integral, t I when a = 9.0, b = 18.7 with t > 1. Give your answer to 3 decimal places. Question 4 5 pts By using implicit differentiation, find the gradient, dy/dx of the tangent to the curve, x² + 4.0y³ - 4.5xy = 8. at the point (2.4,2.29), giving your answer to 3 decimal places. Assume that this point satisfy the given equation of the curve. Question 5 5 pts Consider the differential equation, x-+2y=x2 with the initial condition y(1)=1. Find the value of y when x=3.31. Give your answer to 4 decimal places. Question 6 5 pts Consider the differential equation, dy with the initial condition y(0)=3. Calculate the value of y when x=23.6. Give your answer to 3 decimal places. Question 7 5 pts Consider the differential equation, dy da = y + et sin x with the initial condition y(0)=1. From the particular solution, find the value of y when x=0.80 radian. Give your answer to 4 decimal places. Question 8 5 pts For the following equation, log(y) - log(y+5.94)+ log10.58 = 2log(x) = Determine the value of y when x = 1.40, giving your answer to 4 decimal places. Question 9 5 pts For the following equation, log(6.2-y)-log(y)= log(4.3+x4) - - 5log(x) Determine the value of y when x= 2.1, giving your answer to 3 decimal places. Question 10 5 pts Two complex numbers are given by: W =5+4i z=7+3i = Calculate the imaginery part of w/z. Give your answer to 3 decimal places. Question 11 5 pts A 2nd-order differential equation is given by, a dx² y + b + The complementary function is. = For a=1.39, b=1.53 and c=3.11, find the value of p to 4 decimal places. Question 12 5 pts A 2nd-order differential equation is given by, + b + The complementary function is. yc=e°X(Csin(qx) + Dcos(qx)) For a=1.33,b=0.15 and c=3.88, find the value of q to 3 decimal places. Question 13 5 pts A non-homogeneous 2nd-order differential equation is given by, a dx² d2y + b dx + = The particular integral is, Yp=pe2x Assume no special cases in this problem. For a=0.22, b=8.8 and c=-3.94, find the value of p to 3 decimal places. Question 14 5 pts A non-homogeneous 2nd-order differential equation is given by, a d2y 2 + b dx = The particular integral is, yp=px+q Assume no special cases in this problem. For a=0.57, b=10.9 and c=-3.57, find the value of q to 4 decimal places. Question 15 5 pts Avector p is given by, p = 7.01i + 6.69j + 3.63k Find the magnitude of pto 4 decimal places. Question 16 5 pts Two vectors p and q are given by, p = 3.08i + 1.92j + 4.01k q = 2.81i + 2.23j + 1.19k From the vector cross product pxq. determine the coefficient of the j unit vector to 4 decimal places. Question 17 5 pts Two vectors p and q are given by, p=4.3i+-4.7j q=2i+4j = Calculate the cosine of the angle between vector p and vector q to 3 decimal places. Question 18 5 pts Two vectors p and q are given by, p = - -3.96i + 2.66j q = 3.469i + 1.284j Find the dot product p.q to 4 decimal places. Question 19 5 pts A 2nd-order differential equation is given by, a dx² + = The complementary function is, YC=AeP*+Beax where p>q. For a=2.86,b=10.6 andc=-2.90,findthe value of a to 4 decimal places. Question 20 5 pts A 2nd-order differential equation is given by, The complementary function is, where p> a. For a=0.52, b=9.5 and c=-7.05, find the value of p to 3 decimal places.

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       1          -3.8046
       2          6.0417
       3          37.317...

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