Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Сторінка 16
... third 1 ( 2n - 4 ) ... ( 2 n − r + 1 ) n ( n − 1 ) 2'u ' 1.2 ... ( r - 4 ) 1.2 e1 , and so on . Collecting these terms and multiplying by 1.2 ... r , we find d ' ( u " ) ' u ' = 2n ( 2n − 1 ) ... ( 2n − r + 1 ) un -T nr ( r − 1 ) ...
... third 1 ( 2n - 4 ) ... ( 2 n − r + 1 ) n ( n − 1 ) 2'u ' 1.2 ... ( r - 4 ) 1.2 e1 , and so on . Collecting these terms and multiplying by 1.2 ... r , we find d ' ( u " ) ' u ' = 2n ( 2n − 1 ) ... ( 2n − r + 1 ) un -T nr ( r − 1 ) ...
Сторінка 32
... third time and again multiplying by a + y , du we see that ď u d'u ( a + y ) 3 + 3 ( a + y ) 2 + ( a + y ) dy3 dy dy and therefore d3 u + bu = = 0 . dr = du dx3 " ( 10 ) Transform 1 du du 11 + + = 0 a da dx from a to 0 , having given a2 ...
... third time and again multiplying by a + y , du we see that ď u d'u ( a + y ) 3 + 3 ( a + y ) 2 + ( a + y ) dy3 dy dy and therefore d3 u + bu = = 0 . dr = du dx3 " ( 10 ) Transform 1 du du 11 + + = 0 a da dx from a to 0 , having given a2 ...
Сторінка 48
... third differentiation d'z ď3 z ď z d3 X3 + 3x2y + 3xy + y3 = n ( n − 1 ) ( n − 2 ) ≈ , dax3 dx dy dx dy dy3 See p . 26 . and so on to any order . ( 19 ) Eliminate the functions from the equation ≈ = 4 ( x + at ) + √ ( x − at ) , a ...
... third differentiation d'z ď3 z ď z d3 X3 + 3x2y + 3xy + y3 = n ( n − 1 ) ( n − 2 ) ≈ , dax3 dx dy dx dy dy3 See p . 26 . and so on to any order . ( 19 ) Eliminate the functions from the equation ≈ = 4 ( x + at ) + √ ( x − at ) , a ...
Сторінка 108
... third time on the same supposition , and making a = 0 , y = 0 , dy a = 0 , or = • dx 0 = 0 , we find dx ď y dx2 = 3a Y dy 2 , which being positive shews that = 0 is a minimum . ( 5 ) dy dx = = y1 + x1 - 23 y3 - x , - 4xy + 2 = 0 ...
... third time on the same supposition , and making a = 0 , y = 0 , dy a = 0 , or = • dx 0 = 0 , we find dx ď y dx2 = 3a Y dy 2 , which being positive shews that = 0 is a minimum . ( 5 ) dy dx = = y1 + x1 - 23 y3 - x , - 4xy + 2 = 0 ...
Сторінка 109
... third differentials also vanish , while the values of y ' deduced from the biquadratic d'u +4 dx ' d'u dx3 dy y ' + 6 d1u dx2dy d'u d'u 12 13 • y ̃ + 4 y's + y'1 = 0 , dx dy3 dy1 must be all impossiblet . We may proceed in the same way ...
... third differentials also vanish , while the values of y ' deduced from the biquadratic d'u +4 dx ' d'u dx3 dy y ' + 6 d1u dx2dy d'u d'u 12 13 • y ̃ + 4 y's + y'1 = 0 , dx dy3 dy1 must be all impossiblet . We may proceed in the same way ...
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