An elementary treatise on differential equations and their applications . 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
An elementary treatise on differential equations and their applications . 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. Fig. 2. the tangent could not be parallel to Ox, so it cannot cut it at all, butbecomes asymptotic to it. The other characteristics are of similar nature. Examples for solution. Sketch the characteristics ofdy^dx (1)(2) y(l-x). = x2y. (3) dy dx -f = y + x2dx a 10. Singular points. In all examples like those in the lastarticle, we get one characteristic, and only one, through every point Saj Cut/ of the plane. By tracing the two curves t|=0 and ^-|=0 we caneasily sketch the system. If, however, f(x, y) becomes indeterminate for one or morepoints (called singular points), it is often very difficult to sketch the 8 DIFFERENTIAL EQUATIONS system in the neighbourhood of these points. But the followingexamples can be treated geometrically. In general, a complicatedanalytical treatment is required.* Ex. (i). -j-=-- Here the origin is a singular point. The geo- metrical meaning of the equation is that the radius vector and thetangent have the same gradient, which can only be the case for stra
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