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Assignment1

# Assignment1 - dy dx = e y y 2-2 y Clasify each point as...

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Assignment #1 Due: At the START of tutorial on May 18, 2011. No late assignments will be accepted. 1. Find values of m so that the function y = x m is a solution of the di±erential equation x 2 y ′′ + 2 xy - 6 y = 0 for all x > 0. 2. (a) Verify that the ODE d 2 y dx 2 + 3 dy dx + 2 y = e x has the 2-parameter family of solutions y ( x ) = xe x + c 1 e x + c 2 e 2 x (b) Find the values of c 1 and c 2 for the initial conditions y (0) = 1 and y (0) = 0 3. Consider the ²rst-order ODE x dy dx = 2 y. (1) (a) Show that y ( x ) = Cx 2 satis²es (1). (b) Use the Theorem of Existence and Uniqueness to justify the solutions of this ODE for initial value y ( x 0 ) = y 0 . 4. Find the critical points and draw phase portrait of the autonomous ²rst order ODE
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Unformatted text preview: dy dx = e y ( y 2-2 y ) . Clasify each point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the xy plane determined by the graphs of the equilibrium solutions. 5. Solve the following initial value problems (a) x 2 y ′ = 1-x 2 + y 2-( xy ) 2 with y (1) = 0 (b) (1 + x 2 ) dy dx + 2 xy = f ( x ) where f ( x ) = b x ≤ x < 1-x x ≥ 1 with the initial condition y (0) = 0. 1...
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