where A is a suitably chosen constant, and exp. z denotes e^z. In terms of the two solutions y1, y2 of the differential equation having zero on the right side, the general solution of the equation with R = [phi](x) on the right side can at once be verified to be Ay1 + By2 + y1u - y2v, where u, v respectively denote the integrals _ _ / / u = |y2[phi](x)(y1'y2 - y2'y1)^-1 dx, v = |y1[phi](x)(y1'y2 - y2'y1)^-1 dx. _/ _/
by writing y = v exp. (-½ [int] Pdx), is at once seen to be reduced to d²v/dx² + 1v = 0, where 1 = Q - ½dP/dx - ¼P². If [eta] = - 1/v dv/dx, the equation d²v/dx² + 1v = 0 becomes d[eta]/dx = 1 + [eta]², a non-linear equation of the first order.
where A, B, C are functions of x, is, by the substitution
d²y / 1 dC\ dy --- - ( B + - -- )-- + ACy = 0. dx² \ C dx/ dx
known as Riccati's equation, is transformed into an equation of the same form by a substitution of the form [eta] = (aY + b)/(cY + d), where a, b, c, d are any functions of x, and this fact may be utilized to obtain a solution when A, B, C have special forms; in particular if any particular solution of the equation be known, say [eta]0, the substitution [eta] = [eta]0 - 1/Y enables us at once to obtain the general solution; for instance, when
a particular solution is [eta]0 = [root](-A/C). This is a case of the remark, often useful in practice, that the linear equation
d²y d[phi] dy [phi](x)--- + ½------ -- + [mu]y = 0, dx² dx dx
where [mu] is a constant, is reducible to a standard form by taking a new independent variable _ / z = | dx[[p](x)]^-½. _/
We pass to other types of equations of which the solution can be obtained by rule. We may have cases in which there are two dependent variables, x and y, and one independent variable t, the differential coefficients dx/dt, dy/dt being given as functions of x, y and t. Of such equations a simple case is expressed by the pair
dx dy -- = ax + by + c, -- = a'x + b'y + c', dt dt
wherein the coefficients a, b, c, a', b', c', are constants. To integrate these, form with the constant [lambda] the differential coefficient of z = x + [lambda]y, that is dz/dt = (a + [lambda]a')x + (b + [lambda]b')y + c + [lambda]c', the quantity [lambda] being so chosen that b + [lambda]b' = [lambda](a + [lambda]a'), so that we have dz/dt = (a + [lambda]a')z + c + [lambda]c'; this last equation is at once integrable in the form z(a + [lambda]a') + c + [lambda]c' = Ae^(a + [lambda]a')t, where A is an arbitrary constant. In general, the condition b + [lambda]b' = [lambda](a + [lambda]a') is satisfied by two different values of [lambda], say [lambda]1, [lambda]2; the solutions corresponding to these give the values of x +[lambda]1y and x + [lambda]2y, from which x and y can be found as functions of t, involving two arbitrary constants. If, however, the two roots of the quadratic equation for [lambda] are equal, that is, if (a - b')² + 4a'b = 0, the method described gives only one equation, expressing x + [lambda]y in terms of t; by means of this equation y can be eliminated from dx/dt = ax + by + c, leading to an equation of the form dx/dt = Px + Q + Re^(a + [lambda]a')t, where P, Q, R are constants. The integration of this gives x, and thence y can be found.