An elementary treatise on differential equations and their applications . esented by the Complete Primitive. Illustrate by a graph. 9. Graphical representation. We shall now give some examplesof a method * of sketching rapidly the general form of the family ofcurves representing the Complete Primitive of ?I-• * Due to Dr. S. Brodetsky and Prof. Takeo Wada. DIFFERENTIAL EQUATIONS where f(x, y) is a function of x and y having a perfectly definitefinite value * for every pair of finite values of x and y. The curves of the family are called the characteristics of theequation. dy Ex. (i) Here dx =
An elementary treatise on differential equations and their applications . esented by the Complete Primitive. Illustrate by a graph. 9. Graphical representation. We shall now give some examplesof a method * of sketching rapidly the general form of the family ofcurves representing the Complete Primitive of ?I-• * Due to Dr. S. Brodetsky and Prof. Takeo Wada. DIFFERENTIAL EQUATIONS where f(x, y) is a function of x and y having a perfectly definitefinite value * for every pair of finite values of x and y. The curves of the family are called the characteristics of theequation. dy Ex. (i) Here dx = x(y-l). S^-^l^2^-1)- Now a curve has its concavity upwards when the second differentialcoefficient is positive. Hence the characteristics will be concave upabove y — \, and concave down below this line. The maximum orminimum points he on aj = 0, since dy/dx—O there. The characteristicsnear y = \, which is a member of the family, are flatter than thosefurther from it. These considerations show us that the family is of the general formshown in Fig. 1. y. Fig. 1 Ex. (ii) Here dydx = y + ex. pLjJL+e*=y+2e*. dxi dx We start by tracing the curve of maxima and minima y + ex=0,and the curve of inflexions y + 2ex = 0. Consider the characteristicthrough the origin. At this point both differential coefficients arepositive, so as x increases y increases also, and the curve is concaveupwards. This gives us the right-hand portion of the characteristicmarked 3 in Fig. 2. If we move to the left along this we get to the * Thus excluding a function like y/x, which is indeterminate when x=0 andV=0. GRAPHICAL REPRESENTATION curve of minima. At the point of intersection the tangent is parallel toOx. After this we ascend again, so meeting the curve of crossing this the characteristic becomes convex upwards. It stillascends. Now the figure shows that if it cut the curve of minima again y
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