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Concept 02 / 06Differential Equations

Differential Equations: Order, Degree, Solutions and Formation

What a differential equation and its solution are, order and degree (and when degree is undefined), general versus particular solutions, the number of arbitrary constants, forming a differential equation from a family of curves by eliminating constants, and slope fields.

Figure 1
A differential equation (DE) is an equation involving an unknown function and its derivatives, such as dydx=−xy\frac{dy}{dx} = -\frac{x}{y} or y′′+4y=0y'' + 4y = 0. It expresses a law of change, and solving it recovers the function from that law. Order and degree. The order is the order of the highest derivative present. The degree is the power of that highest-order derivative, once the equation has been made polynomial in all the derivatives (free of radicals and fractional powers in the derivatives). For example, (d2ydx2)3+(dydx)5+y=0\left(\frac{d^{2}y}{dx^{2}}\right)^{3} + \left(\frac{dy}{dx}\right)^{5} + y = 0 has order 22 and degree 33. The equation d2ydx2=[1+(dydx)2]3/2\frac{d^{2}y}{dx^{2}} = \left[1 + \left(\frac{dy}{dx}\right)^{2}\right]^{3/2} must be squared first: (y′′)2=(1+y′2)3\left(y''\right)^{2} = (1 + y'^{2})^{3}, so it has order 22 and degree 22. If the equation contains a derivative inside a transcendental function, such as sin⁡(dydx)\sin\left(\frac{dy}{dx}\right) or ey′e^{y'}, it cannot be written as a polynomial in the derivatives and its degree is not defined. (A term like sin⁡y\sin y or exe^{x} is harmless; only derivatives inside such functions matter.) Solutions. A solution is a function that satisfies the equation on an interval. The general solution of an nnth order equation contains nn independent arbitrary constants; it describes a whole family of curves. A particular solution is obtained by fixing the constants, usually from initial conditions such as y(0)=1y(0) = 1. For dydx=−xy\frac{dy}{dx} = -\frac{x}{y}, the general solution is x2+y2=Cx^{2} + y^{2} = C (circles about the origin), and the condition y(0)=2y(0) = 2 selects x2+y2=4x^{2} + y^{2} = 4. The figure shows a slope field: at each point, a short segment with slope given by the DE. Solution curves are those that run tangent to the segments everywhere. This picture explains why one initial condition picks one curve in a first-order equation: through each point there passes exactly one solution curve (under mild conditions). Forming a DE from a family of curves. Given a family with nn arbitrary constants, differentiate nn times and eliminate the constants between the n+1n + 1 equations. The resulting DE has order nn (equal to the number of essential constants). Examples: y=mxy = mx (lines through the origin, one constant): y′=my' = m, so y=xy′y = xy', order 11. y=Aex+Be−xy = Ae^{x} + Be^{-x} (two constants): y′′=Aex+Be−x=yy'' = Ae^{x} + Be^{-x} = y, so y′′−y=0y'' - y = 0, order 22. Circles of radius rr with centres on the xx-axis, (x−h)2+y2=r2(x - h)^{2} + y^{2} = r^{2} (one constant hh): differentiate to x−h+yy′=0x - h + yy' = 0, so x−h=−yy′x - h = -yy'; substitute: y2y′2+y2=r2y^{2}y'^{2} + y^{2} = r^{2}, order 11, degree 22. Be careful to count only independent constants: y=Aex+By = Ae^{x + B} looks like two constants but is really one (AeBAe^{B}). Verifying a solution: substitute the function and its derivatives into the equation and check that it holds identically. This is also the fastest way to answer multiple-choice questions asking which function solves a given DE. Key Takeaways: - Order: highest derivative; degree: its power after clearing radicals in the derivatives. - Degree is undefined if a derivative appears inside sin⁡\sin, log⁡\log, e(⋅)e^{(\cdot)} and so on. - General solution of order nn has nn independent constants. - Forming a DE: differentiate as many times as there are constants, then eliminate. - Count essential constants: Aex+BAe^{x + B} has only one. - A slope field shows the direction of solution curves at every point. - Pitfall: calculating degree before removing a fractional power of a derivative. Brain-Teaser 1: Find the order and degree of y=xdydx+1+(dydx)2y = x\frac{dy}{dx} + \sqrt{1 + \left(\frac{dy}{dx}\right)^{2}}. Solution: Isolate the root and square: (y−xy′)2=1+y′2(y - xy')^{2} = 1 + y'^{2}. Expanding, y2−2xyy′+x2y′2=1+y′2y^{2} - 2xyy' + x^{2}y'^{2} = 1 + y'^{2}, which is a polynomial in y′y' of degree 22. So the order is 11 and the degree is 22. This is a Clairaut-type equation; its general solution is y=cx+1+c2y = cx + \sqrt{1 + c^{2}}, a family of lines. Brain-Teaser 2: Form the differential equation of all parabolas with axis parallel to the yy-axis. Solution: Such a parabola is y=ax2+bx+cy = ax^{2} + bx + c, with three constants. Differentiate three times: y′=2ax+by' = 2ax + b, y′′=2ay'' = 2a, y′′′=0y''' = 0. So the DE is d3ydx3=0\frac{d^{3}y}{dx^{3}} = 0, of order 33. The number of differentiations is set by the number of constants, and here the constants disappear one per differentiation. Brain-Teaser 3: Form the DE of the family of circles touching the yy-axis at the origin. Solution: Such circles have centre (a,0)(a, 0) and radius ∣a∣|a|: (x−a)2+y2=a2(x - a)^{2} + y^{2} = a^{2}, that is, x2+y2=2axx^{2} + y^{2} = 2ax. One constant. Differentiate: 2x+2yy′=2a2x + 2yy' = 2a. Substitute 2a=2x+2yy′2a = 2x + 2yy' into the original: x2+y2=x(2x+2yy′)x^{2} + y^{2} = x(2x + 2yy'), so y2−x2=2xyy′y^{2} - x^{2} = 2xyy', or dydx=y2−x22xy\frac{dy}{dx} = \frac{y^{2} - x^{2}}{2xy}. This is a homogeneous equation, which is typical of families built from circles and lines through the origin.

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