Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Сторінка x
... 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 ...
Сторінка 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 ...
Сторінка 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 ...
Сторінка 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 ...
Сторінка 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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a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³