HW 1 solutions - HOMEWORK 1 SOLUTIONS Section 1.1#2 3 11...

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HOMEWORK 1 SOLUTIONS Section 1 . 1 : #2 , 3 , 11 , 12 #2) Which of the following operators are linear? (This is accomplished by checking if L u = c L u and L ( u + v ) = L u + L V are satisfied). (a) L u = u x + xu y (b) L u = u x + uu y (c) L u = u x + u 2 y (d) L u = u x + u y + 1 (e) L u = 1 + x 2 cos( y ) u x + u yxy - arctan( x y ) u Answer: Only (a) and (e) are Linear. (b), (c), and (d) do not satisfy L ( cu ) = c L ( u ). #3) For each of the equations below state the order and type. Equation Order Type ( a ) 2 Linear Inhomogeneous ( b ) 2 Linear Homogeneous ( c ) 3 Non-Linear ( d ) 2 Linear Inhomogeneous ( e ) 2 Linear Homogeneous ( f ) 1 Non-Linear ( g ) 1 Linear Homogeneous ( h ) 4 Non-Linear #11 Verify that u ( x, y ) = f ( x ) g ( y ) is a solution for all pairs of differentiable functions f and g of one variable to the equation: uu xy = u x u y Answer: u x = f 0 ( x ) g ( y ) u y = f ( x ) g 0 ( y ) u xy = f 0 ( x ) g 0 ( y ) Then we have: uu xy = f ( x ) g ( y ) f 0 ( x ) g 0 ( y ) = f 0 ( x ) g ( y ) f ( x ) g 0 ( y ) = u x u y #12 Verify by substitution that: u n ( x, y ) = sin( nx ) sinh( ny ) is a solution of u xx + u yy = 0 for every n > 0. Answer: ( u n ) xx = - n 2 sin( nx ) sinh( ny ) and ( u n ) yy = n 2 sin( nx ) sinh( ny ). When we add these together, we get ( u n ) xx + ( u n ) yy = 0.
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