An elementary treatise on differential equations and their applications . ffect the firstseven decimal places, so the required value is Of course for larger values of x we should have to take morethan three approximations to get the result to the required degreeof accuracy. We shall prove in Chap. X. that under certain conditions theapproximations obtained really do tend to a limit, and that this limitgives the solution. This is called an Existence Theorem. Example for solution. (i) Show that in Ex. (ii) of Art. 83, x=0-5 gives y = and2= , while a;=0-2 gives y = l
An elementary treatise on differential equations and their applications . ffect the firstseven decimal places, so the required value is Of course for larger values of x we should have to take morethan three approximations to get the result to the required degreeof accuracy. We shall prove in Chap. X. that under certain conditions theapproximations obtained really do tend to a limit, and that this limitgives the solution. This is called an Existence Theorem. Example for solution. (i) Show that in Ex. (ii) of Art. 83, x=0-5 gives y = and2= , while a;=0-2 gives y = and z = 85. Numerical approximation direct from the differential equation. The method of integrating successive approximations breaks downif, as is often the case, the integrations are impracticable. Butthere are other methods which can always be applied. Considerthe problem geometrically. The differential equation determines a family of curves (the characteristics ) which do notintersect each other and of which one passes through every point. Fig. 23. in the plane.* Given a point P (a, b), we know that the gradientof the characteristic through P is/(a, b), and we want to determine * This is on the assumption that f(x, y) has a perfectly definite value for everypoint in the plane. If, however, f(x, y) becomes indeterminate for one or morepoints, these points are called singular points of the equation, and the behaviourof the characteristics near such points calls for special investigation. See Art. 10. 98 DIFFERENTIAL EQUATIONS the y =NQ of any other point on the same characteristic, given thatx=ON = a + h, say. A first approximation is given by taking thetangent PR instead of the characteristic PQ, taking y =2VX +LR=NL+PL tan /_RPL = b +hf(a, b) =b+hf0, say. But unless h is very small indeed, the error RQ is far fromnegligible. A more reasonable approximation is to take the chord PQ asparallel to the tangent to the characteristic through S, the midd
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