By Raymond Pearl

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1 Definitions and Basic Concepts Directional Field A plot of short line segments drawn at various points in the (x, y) plane showing the slope of the solution curve there is called direction field for the DE. 2 Integral Curves The family of curves in the (x, y) plane, that represent all solutions of DE is called the integral curves. 3 Autonomous Systems The first order DE dy/dx = f (y) is called autonomous, since the independent variable does not appear explicitly. The isoclines are made up of horizonal lines y = m, along which the slope of directional fields is the constant, y = f (m).

52 FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS Moreover, the fundamental theorem says that T is one-to-one (T (y) = T (z) impliesy = z) and onto (every d ∈ Rn is of the form T (y) for some y ∈ V ). A linear transformation which is one-to-one and onto is called an isomorphism. Isomorphic vector spaces have the same properties. 2 Dimension and Basis of Vector Space We call the vector space being n-dimensional with the notation by dim(V ) = n. This means that there exists a sequence of elements: y1 , y2 , .

The above DE can be written in the form a1 (x − x0 ) + b1 (y − y0 ) dy = dx a2 (x − x0 ) + b2 (y − y0 ) which yields the DE a1 X + b1 Y dY = dX a2 X + b2 Y 23 24 FUNDAMENTALS OF ORDINARY DIFFERENTIAL EQUATIONS after the change of variables X = x − x0 , Y = y − y0 . 3 Riccatti equation: y 0 = p(x)y + q(x)y 2 + r(x) Suppose that u = u(x) is a solution of this DE and make the change of variables y = u + 1/v. Then y = u − v /v 2 and the DE becomes u − v /v 2 = p(x)(u + 1/v) + q(x)(u2 + 2u/v + 1/v 2 ) + r(x) = p(x)u + q(x)u2 + r(x) + (p(x) + 2uq(x))/v + q(x)/v 2 from which we get v + (p(x) + 2uq(x))v = −q(x), a linear equation.