Chapter 1: Problems 21 and QxQy + QxyQ = 1 + 48Qxy 3840 (1.13) are equivalent according to the conventions of this textbook. If p(x, y) is a solution to (1.12), then what can we choose for the constants α, β, γ and δ so that Q(x, y) = αp(βx, γy) + δ is a solution to (1.13)? 10. Professor Bogus has carefully studied the ordinary differential equa- tion ff = f and would like to name it after himself. He has seen the following differential equations in papers by other researchers: uu = 5u + 8uu 2uu = 9u + 2u 1 3 uu = u + 4u . Which of them is equivalent to the Bogus Equation according to the conventions of this textbook? 11. Prove that if u(x, t) is a solution to the differential equation ut = uuxx, then so is the function 3u(2x, 12t 7). 12. Consider the equation ut = (ux)2. (1.14) (a) Classify the equation: Is it linear or nonlinear? Partial or or- dinary? Autonomous or nonautonomous? (b) Show that if u(x, t) is a solution to this equation, then so is the function ˆ(x, t) = u(x, t) + γ for any real number γ. (c) What value(s) can be given to the parameter β so that U(x, t) = u(x, βt) is a solution to the equivalent equation Ut + U 2 x = 0 (1.15) whenever u(x, t) is a solution to equation (1.14)? (d) Note that u(x, t) = 3x + 9t is a (simple) solution to (1.14). What is the corresponding solution to (1.15) produced using the procedure from (c)?
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