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Section 2.2 Notes

# Section 2.2 Notes - y = vx 1 dy dx = v x dv dx v x dv dx =...

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Section 2.2: Separable Equations and Equations Reducible to This Form Seperable equations are the simpleset ﬁrst order diﬀerential equa- tions. The equation F ( x ) G ( y ) dx + f ( x ) g ( y ) = 0 can be multiplied by the integrating factor 1 f ( x ) G ( y ) . In doing this we separate the variables and simple integration can solve the diﬀeren- tial equation. However, some solutions may have been lost in the process. So, the missing solutions can be found by solving G ( y ) = 0. Any solution the said equation is a solution of the original diﬀeren- tial equation which may or may not have been lost in the separation process. Deﬁnition 1 The diﬀerential equation M ( x,y ) dx + N ( x,y ) dy = 0 is homogeneous if when written in the form dy dx = f ( x,y ) , there exists a function g such that f ( x,y ) can be expressed in the form g ( y/x ) Theorem 1 If M ( x,y ) dx + N ( x,y ) dy = 0 is a homogeneous equation, then the change of variables y = vx transforms into a seperable equation. Proof: Since M ( x,y ) dx + N ( x,y ) dy = 0 is a homogeneous, it may be written in the form dy dx = g ( y x ) Let

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Unformatted text preview: y = vx 1 dy dx = v + x dv dx v + x dv dx = g ( v ) Seperating the variables dv v-g ( v ) + dx x = 0 Z dv v-g ( v ) + Z dx x = c Let F ( v ) = Z dv v-g ( v ) Then, the solution takes the form F ( y x ) + ln | x | = c Example 1 Solve the equation ( x 2-3 y 2 ) dx + (2 xy ) dy = 0 dy dx = 3 y 2-x 2 2 xy 3 y 2-x 2 2 xy = 3 y 2 x-x 2 y = 3 2 ( y x )-1 2 ( 1 y x ) dy dx = 3 2 ( y x )-1 2 ( 1 y x ) The right part is of the form g ( y x ) . So, the diﬀerential equation is homogeneous. We can now proceed to solve the equation. ( x 2-3 y 2 ) dx + (2 xy ) dy = 0 dy dx = 3 y 2 x-x 2 y 2 Let y = vx v + x dv dx =-1 2 v + 3 v 2 x dv dx =-1 2 v + v 2 x dv dx = v 2-1 2 v Seperate the variables 2 v v 2-1 dv = dx x ln ± ± v 2-1 ± ± = ln | x | + ln | c | ± ± v 2-1 ± ± = | cx | Replace v by y x ± ± ± ± y 2 x 2-1 ± ± ± ± = | cx | y 2-x 2 = cx 3 3...
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Section 2.2 Notes - y = vx 1 dy dx = v x dv dx v x dv dx =...

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