Ordinary Diff Eq Exam Review Solutions 11

Ordinary Diff Eq Exam Review Solutions 11 - 1 Solutions the...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Solutions the equation or we can solve explicitly to get where the negative square root is used since . 11. In standard form we get . Clearly, solutions. Separating the variables gives 13 , are equilibrium Integrating both sides of this equation (using the substitution for the integral on the left) gives , Multiplying by , taking the exponential of both sides, and removing the absolute values gives where is a nonzero constant. However, when the equation becomes and hence . By considering an arbitrary constant (which we will call ), the implicit equation includes the two equilibrium solutions for . 12. The variables are already separated, so integrate both sides to get , a real constant. This can be simplified to . (where we replace by ) We leave the answer in implicit form. 13. The variables are already separated, so integrate both sides to get , a real constant. Simplifying gives leave the answer in implicit form . We 14. There is an equilibrium solution . Separating variables give and integrating gives . Thus , a real constant. This is equivalent to writing , a real constant, since twice an arbitrary constant is still an arbitrary constant. 15. In standard form we get so is a solution. Separating variables gives . The function is continuous on the interval and so has an antiderivative. Integration gives . Multiplying by and exponentiating gives where is a positive constant. Removing the absolute value signs gives , with . If we allow we get the equilibrium solution . Thus the solution can be written , any real constant. 16. An equilibrium solution is . Separating variables gives , a real constant. Simplifying and integrating gives gives , and the equilibrium solution . ...
View Full Document

Ask a homework question - tutors are online