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2.1-2.2 - Sections 2.1-2.2 Integrating factor p(x)y = g(x...

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Sections 2.1-2.2 Integrating factor dy dx + p ( x ) y = g ( x ) integrating factor μ ( x ) Step 1: Your original equation may not be in this form. Recognize p ( x ) and q ( x ) after transformation. No sign mistakes!! Step 2: Set μ ( x ) = exp p ( x ) dx . This is an indefinite integral, but we usually choose the arbitrary constant c to be 0 at this step for convenience. Step 3: New equation μ ( x ) y = μ ( x ) g ( x ) dx + c . Solve for y . Step 4: For initial value problem, if you can not find an explicit solution for y , then the best thing to do is to set the lower limit of integration to be the x -coordinate of the given point, as in Example 4. Remark : You will constantly be asked to determine the interval of existence for your solution. Example 3, a better way to see this is by looking at the given point (1 , 2), the x -coodinate 1 is to the positive y -axis. Separation separable form M ( x ) dx + N ( y ) dy = 0 separation Step 1: Your original equation may not be in this form. Recognize M ( x ) and N ( y ) after transformation. No sign mistakes!!

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