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AM3815-Test1-Solns

AM3815-Test1-Solns - AM 3815 Solutions to Test 1 1(a...

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AM 3815 - Solutions to Test 1 1. (a) non-linear, (b) linear, (c) non-linear, (d) linear. 2. (a) Characteristic curves are [ x ( t ), y ( t )] = [2 t , - 3 t + s ] for varying values of s , so the appro- priate sketch is given below. (b) Disagree. Characteristic curves of the equation f ( x , y ) u x + g ( x , y ) u y = F ( u , x , y ) follow the direction u · [ f , g ]. For these to be level sets, F would have to be zero, but clearly this is not always the case. 3. (a) The equation can be written as ( x + y )( x - y ) u = 0 and so solutions are u ( x , y ) = f ( y - x ) + g ( y + x ), or u ( x , y ) = F ( x - y ) + G ( x + y ). *There are alternative approaches that require more steps. A correct solution via those methods is still only worth 2 marks total (b) If we write u · [ y , - x ] = 0, then we know to solve dy dx = - x y to find y 2 = - x 2 + C . So, C = y 2 + x 2 is the appropriate “composite” variable and we have u ( x , y ) = f ( y 2 + x 2 ). *There are alternative approaches that require more steps. A correct solution via those methods is still only worth 2 marks total.
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(c) Define v = u x and restate the equation, v y + v = x . Solve this new equation (e.g. using
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