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JHU 553

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JHU 553.291, Fall 2021 – Homework #12 Slope fields, 2nd and nth order linear D.E.s, Euler’s Method for 1st order D.E.s Due on on Friday 11/19 (will accept until Monday 11/29 by 11:59pm), on Gradescope 
Instructions: If your homework is not already in electronic format, please scan it into one PDF file, and upload it on GradeScope. Possible apps for scanning your work are CamScanner or GeniusScan: if you believe that turning your document into Black & White (not grayscale!) makes it more readable, please do so (also: writing in pen instead of pencil may make the scanned document more readable). GradeScope makes it possible to ‘tag’ the individual problems within your PDF file: please do it in order to make the graders’ work a bit simpler. Please solve the problems in the given order, and show your work. 
Remember: The general solution to a D.E. depends on arbitrary constants; for a particular solution, the values of such constants are determined by initial conditions (IC) or boundary conditions (BC). 
Problem 12.1 (Slope fields.). Consider the first-order (nonlinear!) differential equation: 
y y' + x = 0 
(a) Rewrite it in the form y' = f(x, y), and plot, by hand, the slope field f(x, y) at the points P1 = (0, 1), P2 = (1, 1), P3 = (1, 0), P4 = (1,—1), P5 = (0,—1), P6 = (-1,—1), P7 = (-1,0), P8 = (-1,1). (All you have to do is to plot short segments at the given points, with slope given by the value of the function f at those points. If at a point (x, y) you haeve f(x, y) = ±1 because the denominator is zero while the numerator is nonzero, just draw a short vertical segment.) 
(b) For any differential equation of the form y' = f(x,y), the slope field, at each point (x0, y0) of the xy-plane, is parallel to the vector 
f (x0, y0) (why?). Show that in the specific case of the equation yy' + x = 0, at any point (x0, y0) of the xy-plane the vector v (and therefore the slope field!) is orthogonal to the “radial direction” vector r = Ix0] . This should be consistent with your findings of part (a). y0 (c) Finally, solve the differential equation (it is separable). Write the general solution (that depends on a constant C) in both implicit and explicit form. For any choice of the constant C, the graph of the solution is a very well-known curve: what is it? What is the significance of the constant C? (d) Choose three different values of the constant C, and draw the graphs of the corresponding solutions on the xy-plane. Verify that the graphs are tangent to the slope field. 
v = 

Problem 12.2. Find the general solutions y(x) of the following second-order homogeneous linear D.E.s: (a) y'' = 0 (b) y'' + 3y = 0 (c) 2y'' + 5y' + 2y = 0 (d) y'' - 4y' + 4y = 0 Problem 12.3. Find the particular solutions y(x), meeting the given initial conditions, of the following second-order homogeneous linear differential equations: 
(a) y'' — y = 0, y(0) = 2, y'(0) = 3 (b) 5y'' + 8y' + 5y = 0, y(0) = 0, y'(0) = 1 
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