Examples of the Processes of the Differential and Integral Calculus |
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... Singular Solutions of Differential Equations 400 IX . Quadrature of Areas and Surfaces , Rectification of Curves , and Cubature of Solids 412 X. Geometrical Problems Involving the Solution of Differential Equations 440 XI . Evaluation ...
... Singular Solutions of Differential Equations 400 IX . Quadrature of Areas and Surfaces , Rectification of Curves , and Cubature of Solids 412 X. Geometrical Problems Involving the Solution of Differential Equations 440 XI . Evaluation ...
Page 331
... singular solution is found by eliminating p between the given equation and x + ƒ ̃ ( p ) = 0 . This equation is ... singular solution is ( 11 ) Let x2 + y2 = n2 . y = px + p - p2 . The general integral is y = C ( x + 1 − C ) ; the ...
... singular solution is found by eliminating p between the given equation and x + ƒ ̃ ( p ) = 0 . This equation is ... singular solution is ( 11 ) Let x2 + y2 = n2 . y = px + p - p2 . The general integral is y = C ( x + 1 − C ) ; the ...
Page 332
... singular solution is dy 2 ( 15 ) Let axy 2 dx 1. dx The general integral is a C ( c- c2 ) . 4ay + ( 2x - b ) = 0 . + ( ba ay - ab ) - dy dr - bay = 0 . On multiplying by 4xy , and taking a and y as the new variables , the equation ...
... singular solution is dy 2 ( 15 ) Let axy 2 dx 1. dx The general integral is a C ( c- c2 ) . 4ay + ( 2x - b ) = 0 . + ( ba ay - ab ) - dy dr - bay = 0 . On multiplying by 4xy , and taking a and y as the new variables , the equation ...
Page 334
... singular solution y = Cx . Sometimes an equation may be considered homogeneous by reckoning a as of one dimension , y of n dimensions , and d2y da - consequently of ( n - 1 ) dimensions , and of ( n = 2 ) dy da = x - 1t , dimensions ...
... singular solution y = Cx . Sometimes an equation may be considered homogeneous by reckoning a as of one dimension , y of n dimensions , and d2y da - consequently of ( n - 1 ) dimensions , and of ( n = 2 ) dy da = x - 1t , dimensions ...
Page 335
... singular solution . y = Ca2 , dy d2y If be reckoned of o dimensions so that y ' d are of the same dimensions , a homogeneous equation may be integrated by assuming dy = dy y p = uy , q = vy ; whence as uydr and udy + ydu = vydx , = udx ...
... singular solution . y = Ca2 , dy d2y If be reckoned of o dimensions so that y ' d are of the same dimensions , a homogeneous equation may be integrated by assuming dy = dy y p = uy , q = vy ; whence as uydr and udy + ydu = vydx , = udx ...
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