math-4b-cauchy-euler34764e76c2e465cc8cb6ff0000369d41.pptx

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Differential Equations Cauchy-Euler Equations Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB

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A Cauchy-Euler equation is a specific type of D.E. that can be put in the form: Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB To solve this we will make this variable substitution: This one is 2 nd -order, but the solution method we will see generalizes to higher order equations as well. In the original equation, y is a function of t. When we substitue, we are thinking of y as a function of x, which in turn is a function of t. So it is very much like solving an integral by substitution, where you choose an intermediate variable that simplifies the integrand. It will help if we use the Leibniz notation for our derivatives. Our equation is: ) t ( g y y t y t 2 x e t ) t ln( x ) t ( g y dt dy t dt y d t 2 2 2
Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB We need to find the derivatives with the chain rule, since we now have a function y(x(t)): We are using this substitution: t 1 dt dx ) t ln( x ) t ( g y dt dy t dt y d t 2 2 2 t 1 dx dy dt dx dx dy dt dy 2 2 2 2 2 2 2 2 2 2 2 2 2 t 1 dx dy t 1 dx y d dt y d t 1 dx dy t 1 dt dx dx y d dt y d t 1 dt d dx dy t 1 dx dy dt d t 1 dx dy dt d dt y d

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Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Next, plug in our expressions to get the equation in terms of x rather than t:

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