SOLVING FIRST ORDER DIFFERENTIAL EQUATIONS Methods for solving Linear, Exact, Separable, Homogeneous, and Bernoulli types Linear 1. Rewrite in the form dy dx x y x + = ( ) ( ) terms of terms of 2. Determine the integrating factor e P x dx ( ) ∫ , where P ( x ) is the factor multiplied by y above. 3. Multiply the equation by the integrating factor. 4. Discard the center term of the expression. (This happens by integrating and taking the derivative.) 5. Rewrite the left term of the equation in the form d dx x y ( ) new terms of . 6. Integrate the equation with respect to x . Do this by removing the d dx from the left side and adding the dx ∫ notation to the right. There will be a constant of integration in the result. 7. Solve for y x ( ) . Exact 1. Rewrite in the form ( ) ( ) terms of and terms of and x y dx x y dy + =0 that is: Mdx Ndy + =0 . 2. The equation is exact if the partial derivative of M with respect to y equals the partial derivative of N with respect to x ; ¶ ¶ ¶ ¶ M y
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