Math 310 Q4 Sol

# Math 310 Q4 Sol - MATH 310 Fall 2011 Differential Equations...

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MATH 310, Fall 2011: Diﬀerential Equations Simon Fraser University Quiz 4: Solutions 1. Consider the diﬀerential equation (of Bernoulli type) t 2 y 0 = 2 ty - t 5 e t y ( t > 0 , y > 0) . Show that using the substitution v = y 3 / 2 , this DE reduces to a linear diﬀerential equation for v ( t ). Write the resulting linear DE for v in standard form. [You should not attempt to solve the diﬀerential equation.] Solution: Solving for y 0 = dy/dt , the diﬀerential equation can be written in the form dy dt = 2 t y - t 3 e t y - 1 / 2 , which is a Bernoulli equation (with n = - 1 / 2 ), for which the appropriate substitution, as given, is v = y 1 - n = y 1 - ( - 1 / 2) = y 3 / 2 . Writing v = y 3 / 2 , we need to ﬁnd the diﬀerential equation for v ( t ) . Diﬀerentiating and using the chain rule, v 0 = dv dt = d dt y 3 / 2 = 3 2 y 1 / 2 dy dt ; now we can substitute for dy/dt from the diﬀerential equation, to ﬁnd dv dt = 3 2 y 1 / 2 ± 2 t y - t 3 e t y - 1 / 2 ² = 3 t y 3 / 2 - 3 2 t 3 e t .

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## This note was uploaded on 01/30/2012 for the course MATH 310 303 taught by Professor Rwk during the Spring '11 term at Simon Fraser.

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Math 310 Q4 Sol - MATH 310 Fall 2011 Differential Equations...

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