By Samuel Zaidman
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Additional info for Abstract differential equations
T/ are linearly independent. t/ D e r2 t are linearly independent for any two real numbers r1 ¤ r2 ¤ 0. t/ ¤ 0 for all t in R. t/ are linearly independent. t/ D e rt . t/ are linearly independent. e rt C rte rt / e 2rt ¤ 0; for all t. t/ are linearly independent. t/ are linearly dependent functions, then their Wronskian must be zero at every point t. 3 The following theorem applies even for the second order differential equations with variable coefficients. 7) where p and q are continuous on an open interval I .
48) This implies d . 49) d . 50). 41), do the following: 1. 47), (the p q from). 2. x/dx . 3. 51). 4. 52). x/ D x. x/ D x 2 . x/ D 12 x 2 . 0; =2/. 0; =2/. 61) f 1; 1g. 0; C1/. 12 Solve in I D . 66) q form. 1. x=2/ C C ; cos x C D c2 c1 : Substitutions and Transformations In this section, we introduce some substitutions and transformations tools to solve some nonlinear first order differential equations. This includes the homogeneous equations and Bernoulli’s equation. y=x/. 68) can be expressed as a function of y=x.
X; y/dy D 0; is called an exact equation. x; y/ D xy. 31) is not always obvious. So, we need a test for exactness and a method for finding the function f . Thus, we have the following theorem. x; y/ 2 R. 32) is a necessary condition for exactness. x; y/ are equal if they are continuous. 32). 31) is also a sufficient condition for exactness. 31). y/ since it must disappear under the derivative with respect to x. 31). 35) Now, we need to verify that the above integrand is a function of y only. 32).
Abstract differential equations by Samuel Zaidman