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77_pdfsam_math 54 differential equation solutions odd

77_pdfsam_math 54 differential equation solutions odd -...

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Exercises 2.5 equation has an integrating factor depending on y alone. Also, since ∂M/∂y ∂N/∂x N = (6 y 2 + 4 y ) (3 y 2 + 2 y ) 3 y 2 x + 2 xy = 3 y 2 + 2 y x (3 y 2 + 2 y ) = 1 x , the equation has an integrating factor depending on x . Writing the equation in the form dx dy = 3 y 2 x + 2 xy 2 y 3 + y 2 = xy (3 y + 2) 2 y 2 ( y + 1) = y (3 y + 2) 2 y 2 ( y + 1) x we conclude that it is separable and linear with x as the dependent variable. 3. This equation is not separable because of the factor ( y 2 + 2 xy ). It is not linear because of the factor y 2 . To see if it is exact, we compute M y ( x, y ) and N x ( x, y ), and see that M y ( x, y )2 y + 2 x = 2 x = N x ( x, y ) . Therefore, the equation is not exact. To see if we can find an integrating factor of the form µ ( x ), we compute ∂M ∂y ∂N ∂x N = 2 y + 4 x x 2 , which is not a function of x alone. To see if we can find an integrating factor of the form
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