Math 225 Differential Equations Notes Chapter 2

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2.2.1 Method for Solving Separable <strong>Equations</strong> Start with: dy dx = g(x)p(y) multiply by dx and by h(y) := 1/p(y) to obtain Then integrate both sides: ∫ h(y)dy = g(x)dx ∫ h(y)dy = g(x)dx Obtain the implicit solution of the differential equation by merging together the integration constants. H(y) = G(x) + C 2
Example 1: Solve the non-linear equation Solution: 1. y 2 dy = (x − 5)dx 2. ∫ y 2 dy = ∫ (x − 5)dx dy dx = x − 5 y 2 3. y3 3 = x2 2 − 5x + C ( ) 1/3 3x 4. y = 2 2 − 15x + 3C ( ) 1/3 3x 5. y = 2 2 − 15x + K 3
Page 1: Math 225 Differential Equations Not
Page 5 and 6: 2.2.2 Formal Justification of Metho
Page 7 and 8: A linear differential equation is s
Page 9 and 10: Example 3: Find the general solutio
Page 11 and 12: To find C we need to apply the init
Page 13 and 14: 2.4 Exact Equations It is also poss
Page 15 and 16: Definition Exact Differential Form
Page 17 and 18: 2.4.1 Method for Solving Exact Equa
Page 19 and 20: 2.5 Exact Equations Can we use inte
Page 21 and 22: Definition Homogeneous Equations If
Page 23 and 24: 2.6.1 Equations of the form dy dx I
Page 25 and 26: To solve Bernoulli equations, we ca

Math 225 Differential Equations Notes Chapter 2

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