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# set2 - \$ Set 2 First Order ODEs Part 1 Kyle A Gallivan...

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a39 a38 a36 a37 Set 2: First Order ODEs - Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Ordinary Differential Equations Fall 2009 1

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a39 a38 a36 a37 First Order ODE Initial Value Problem t R mapsto→ y ( t ) R ( t, y ) R 2 mapsto→ f ( t, y ) R y ( t ) is the highest order of derivative involved The initial value problem: y = f ( t, y ) , y ( t 0 ) = y 0 no general solution so we consider subclasses: linear equations separable equations exact equations 2
a39 a38 a36 a37 Undetermined Coefficients The Undetermined Coefficients Method: hypothesizes a form of the solution that is a function of a set of coefficients substitutes the form of the solution into the differential equation solves for the unknown coefficients if possible This will be used later on more complicated problems but we apply it to a simple problem here. 3

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a39 a38 a36 a37 Undetermined Coefficients Consider the general first order linear constant-coefficient ODE y = ay b (1) where a negationslash = 0 and b are real constants. if b = 0 we know y 1 = ay 1 y 1 ( t ) = Ce at Equation (1) adds a constant to y 1 to get y Equation (1) says y proportional to y Hypothesize that there is a constant k such that y ( t ) = y 1 ( t ) + k 4
a39 a38 a36 a37 Undetermined Coefficients substitute y ( t ) = y 1 ( t ) + k into ODE RHS : ay b = ay 1 ( t ) + ak b = aCe at + ak b substitute y ( t ) = y 1 ( t ) + k into ODE LHS : dy dt = d dt ( y 1 ( t ) + k ) = d dt ( Ce at ) + d dt ( k ) = aCe at equate RHS and LHS to get conditions on k and check for consistency with assumptions: if aCe at = aCe at + ak b then 0 = ak b k = b a 5

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a39 a38 a36 a37 Undetermined Coefficients Since a negationslash = 0 , k is a well-defined constant.
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