Rotz Theorem known as the Principle of Superposition Consider the secondorder linear homogeneous ordinary di64256erential equation 00 0 If and are both solutions to then for any two constants and is also a solution to Proof The fa ID: 26635 Download Pdf

Rotz Theorem known as the Principle of Superposition Consider the secondorder linear homogeneous ordinary di64256erential equation 00 0 If and are both solutions to then for any two constants and is also a solution to Proof The fa

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MA 222 Proof of the Principle of Superposition K. Rotz Theorem known as the Principle of Superposition ): Consider the second-order, linear, homogeneous ordinary diﬀerential equation 00 ) + ) + ) = 0 If and are both solutions to ( ), then for any two constants and is also a solution to ( ). Proof : The fact that and are solutions to ( ) imply that 00 ) + ) + ) = 0 and (1) 00 ) + ) + ) = 0 (2) Since and are constants, we have ) = ) + and 00 ) = 00 ) + Inserting these into ( ), we see that 00 ) + ) + ) = )( 00 }| 00 ) + )) + )( }| ) + )) )( ) + {z ) (3) We now regroup the

terms in (3) by those terms with ’s and ’s: 00 ) + ) + ) = 00 ) + ) + ) + ) + 00 ) + ) + )] 00 ) + ) + )] (4) By equations (1) and (2), the right-hand side of (4) is zero. In other words, 00 ) + ) + ) = 0 so that is a solution to ( ).

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