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MAT 3237 Differential Equations

MAT 3237 Differential Equations. Section 2.5 Solutions by Substitutions Part I. http://myhome.spu.edu/lauw. Quiz. 2.4, 2.5 Part I. Expectations on Presentations. Pay attention to the transitional statements. They are part of the complete solutions. Recall.

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MAT 3237 Differential Equations

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  1. MAT 3237Differential Equations Section 2.5 Solutions by Substitutions Part I http://myhome.spu.edu/lauw

  2. Quiz • 2.4, 2.5 Part I

  3. Expectations on Presentations • Pay attention to the transitional statements. • They are part of the complete solutions.

  4. Recall So far, we know how to solve three types of first order D.E. Two of them are: • Separable D.E. • Linear D.E.

  5. Rationale of Solutions by Substitutions D.E. NOT in the forms above General Solutions

  6. Rationale of Solutions by Substitutions D.E. in the forms above (new variable) Substitution D.E. NOT in the forms above General Solutions

  7. Rationale of Solutions by Substitutions D.E. in the forms above (new variable) Substitution D.E. NOT in the forms above “Backward” Substitution Solve for the solutions (new variable) General Solutions

  8. Preview • Part I • Homogeneous Equations • Part II • Bernoulli’s Equation • “Linear Polynomial Reduction”

  9. Preview • Part I • Homogeneous Equations • ho·mo·ge·ne·ous or ho·mog·e·nous

  10. Definition A first order D.E. is homogeneous if

  11. Definition A first order D.E. is homogeneous if “uniform”…

  12. Fact First order homo. D.E. can be solved (theoretically) by the substitutions (Separable Equations)

  13. Expectations • Precise arguments are very important here. • You need to explain your arguments in (verbally /mathematically) carefully. • A sample is printed on the handout and classwork.

  14. Example 1 • Show that the D.E. is homog. • Use a substitution to solve the D.E.

  15. Example 1 • Show that the D.E. is homo.

  16. Example 1 2. Use a substitution to solve the D.E.

  17. Check Point • You should get a separable equation.

  18. Rationale D.E. in the forms above Substitution D.E. NOT in the forms above “Backward” Substitution Solve for the solutions (new variable) General Solutions

  19. Review: We learned… • the general idea of solutions by substitution. • how to identify homogeneous DE. • the substitution(s) that will transform the DE into separable DE

  20. Next • Section 2.5 Part II

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