Math 310 Q4 Sol - MATH 310, Fall 2011: Differential...

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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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Math 310 Q4 Sol - MATH 310, Fall 2011: Differential...

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