homogeneous_PS

homogeneous_PS - GE 207K Homogenous Equations Problem Set...

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GE 207K Homogenous Equations – Problem Set September 14, 2010 Solve the following ﬁrst-order diﬀerential equations. 1. ( x 2 + y 2 ) dx - 2 xydy . Rewrite the equation as 2 xy dy dx - ( x 2 + y 2 ) = 0 . Divide through by x 2 : 2 ± y x ² y 0 - ³ 1 + ± y x ² 2 ´ = 0 (1) Perform the following substititions u = y x y = ux dy dx = u + u 0 x Therefore, 2 u ( u + u 0 x ) - ( 1 + u 2 ) = 0 Simplifying, we have u + u 0 x = 1 2 ³ 1 u + u ´ u 0 x = - u 2 - 1 2 u Now, we can clearly see we have a separable equation. Solve by separating the variables on either sides of the equation and integrating: - Z 2 u u 2 - 1 du = Z 1 x dx Therefore, - ln | u 2 - 1 | = ln x + C 1 1 u 2 - 1 = C 2 x Don’t forget the back-substitute u = y x . The problem statement was in terms of y and x , and so should be the solution! 1 ( y x ) 2 - 1 = C 2 x Beatifying the solution, we get y 2 - x 2 = C 3 x . (2) 1

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GE 207K Homogenous Equations – Problem Set September 14, 2010 2. Solve the diﬀerential equation (3 x + y ) dx + xdy
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homogeneous_PS - GE 207K Homogenous Equations Problem Set...

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