Question

(Generalized Riccati Equation) Let po, Pi, Pp2 T-R be continous functions defined on an interval I of R. Then the 1st-order differential equations of the type is called generalized Riccati equations. It is another nonlinear ordinary differential equation (a) Suppose, P2 differentiable and P2メ0 on I. By using the Ansatz u(z) :-y(r) P2(x) T, for every z where y is a solution of (2), develope a method to solve the equation (2). Describe in brief steps your method. Hint: The Ansatz (3) is different to the one in Trench, but you might find it in other literature. Do some research! (b) Find all solutions to the following two differential equations. Use Mathematica to plot a direction field to each of the differential equations, and plot in the same diagramm a few solutions of each differential equations. Provide the Mathematica-code and the plot generated by Mathematica. (i) y' + (2)yry (ii) y, = e2"t (1-2e)y+92

Answer 1

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Mathematica Mon May 27, 7:28 AM Untitled-2.nb-Wolfram Mathematica 11.3 Eile Eet sert Format Cell graphics Faluatien Palettes Windam Hep (i) Given differential equation is - e2* +(1-2e*) yy2 dx 114-p3x StreamPlot [ {1, Exp [2+x) + (1-2 * Exp [x)) *y + y ^ 2), (X,-10, 10), {y,-10, 10)] 2. -5 -10 -5 10 12e TanhArcTanh(1 2 ye] 150%-

Mathematica Mon May 27, 7:28 AM Untitled-2.nb-Wolfram Mathematica 11.3 Eile Eet sert Format Cell graphics Faluatien Palettes Windam Hep Hence solution of given differential equation is -1 + 2 ex-Tanhị 측 +ArcTanh(1-2 yol) where y(0) = yo na-sol4-yex]/.sol3t1, 1] Oullel- 0 부 labels image size, background' axes, mere..

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