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Assignment2

# Assignment2 - substitution y x = y 1 x u x you can...

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Assignment #2 Due: At the START of tutorial on June 1, 2011. No late assignments will be accepted. 1. Solve the following diﬀerential equations (a) x ln( x ) dy dx + y = 2 ln x (b) yy 0 = xe y/x + y 2 x (c) (2 x + y + 1) dy dx = 1 (d) ydx + (2 xy - e - 2 y ) dy = 0 2. (a) Show that we can transform the diﬀerential equation x dy dx + P ( x ) y = Q ( x ) y ln y into a linear diﬀerential equation by using the substitution u ( x ) = ln y . (b) Use this method to solve x dy dx = 2 x 2 y + y ln y 3. A Riccati equation is a diﬀerential equation with the form y 0 = p ( x ) + q ( x ) y + r ( x ) y 2 (a) Suppose we are able to ﬁnd a particular solution y 1 ( x ). Show that by using the
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Unformatted text preview: substitution y ( x ) = y 1 ( x )+ u ( x ) you can transform a Riccati equation into a Bernoulli’s equation. (b) Find one-parameter family of solutions for the diﬀerential equation dy dx + 2 xy = 1 + x 2 + y 2 knowing that y 1 ( x ) = x is a solution. 4. (a) Find an implicit solution of IVP dy dx = y 2-x 2 xy , y (1) = √ 2 . (b) Write the solution explicitly and give the largest interval I over which the solution is deﬁned. 1...
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