de-t1-a(1)

# The substitution allows us to reduce b to a linear

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The substitution allows us to reduce (b) to a linear ODE in v and x , namely . Solving this and substituting back results in a 1-parameter family of implicit solutions: . Getting an explicit solution now is cheap thrills. ______________________________________________________________________ Bonus Noise : The substitution v = y converts the ODE into the first order linear equation (*) Using the integrating factor µ =e x leads to the explicit solution to (*) . Finally, .

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TEST1/MAP2302 Page 2 of 3 ______________________________________________________________________ (c) Linear as written. Near θ = 0, an integrating factor is easy to come by: for θε (- π /2, π /2). Multiplying both sides of the DE by µ results in the following derivative equation: By integrating, we have Applying the initial condition, it follows thatC=4 . Hence, an explicit solution near θ = 0 is given by (d) After minimal algebra, this may be seen as either exact with an easy solution, or homogeneous , of degree 1, with a messier solution. As Exact : The usual prestidigitation leads to A 1-parameter family of solutions: As Homogeneous: The degree of homogeneity is 1. Thus, write the
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The substitution allows us to reduce b to a linear ODE in v...

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