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First Order Differntial Equation document sample
PROGRAMME 24 FIRST-ORDER DIFFERENTIAL EQUATIONS STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Introduction A differential equation is a relationship between an independent variable x, a dependent variable y and one or more derivatives of y with respect to x. The order of a differential equation is given by the highest derivative involved. dy x y 2 0 is an equation of the 1st order dx d2y xy 2 y 2 sin x 0 is an equation of the 2nd order dx d3y dy 3 y e 4 x 0 is an equation of the 3rd order dx dx STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Formation of differential equations Differential equations may be formed from a consideration of the physical problems to which they refer. Mathematically, they can occur when arbitrary constants are eliminated from a given function. For example, let: dy y A sin x B cos x so that A cos x B sin x therefore dx d2y 2 A sin x B cos x y dx d2y That is y0 dx 2 STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Formation of differential equations Here the given function had two arbitrary constants: y Asin x B cos x and the end result was a second order differential equation: d2y 2 y0 dx In general an nth order differential equation will result from consideration of a function with n arbitrary constants. STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Introduction Direct integration Separating the variables Homogeneous equations – by substituting y = vx Linear equations – use of integrating factor STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Introduction Solving a differential equation is the reverse process to the one just considered. To solve a differential equation a function has to be found for which the equation holds true. The solution will contain a number of arbitrary constants – the number equalling the order of the differential equation. In this Programme, first-order differential equations are considered. STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Direct integration If the differential equation to be solved can be arranged in the form: dy f ( x) dx the solution can be found by direct integration. That is: y f ( x)dx STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Direct integration For example: dy 3x 2 6 x 5 dx so that: y (3 x 2 6 x 5)dx x3 3x 2 5 x C This is the general solution (or primitive) of the differential equation. If a value of y is given for a specific value of x then a value for C can be found. This would then be a particular solution of the differential equation. STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Separating the variables If a differential equation is of the form: dy f ( x) dx F ( y ) Then, after some manipulation, the solution can be found by direct integration. F ( y )dy f ( x)dx so F ( y)dy f ( x)dx STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Separating the variables For example: dy 2x dx y 1 so that: ( y 1)dy 2 xdx so ( y 1)dy 2 xdx That is: y 2 y C1 x 2 C2 Finally: y 2 y x2 C STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Homogeneous equations – by substituting y = vx In a homogeneous differential equation the total degree in x and y for the terms involved is the same. For example, in the differential equation: dy x 3 y dx 2x the terms in x and y are both of degree 1. To solve this equation requires a change of variable using the equation: y v( x ) x STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Homogeneous equations – by substituting y = vx To solve: dy x 3 y dx 2x let y v( x ) x to yield: dy dv x 3 y 1 3v v x and dx dx 2x 2 That is: dv 1 v x dx 2 which can now be solved using the separation of variables method. STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Linear equations – use of integrating factor Consider the equation: dy 5 y e2 x dx Multiply both sides by e5x to give: e5 x dy dx e5 x 5 y e5 x e2 x that is d dx ye5 x e7 x then: d ye e dx so that ye5 x e7 x C 5x 7x That is: y e 2 x Ce 5 x STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Solution of differential equations Linear equations – use of integrating factor The multiplicative factor e5x that permits the equation to be solved is called the integrating factor and the method of solution applies to equations of the form: Py Q where e dy Pdx is the integrating factor dx The solution is then given as: y.IF Q.IFdx where IF e Pdx STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Introduction Formation of differential equations Solution of differential equations Bernoulli’s equation STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Bernoulli’s equation A Bernoulli equation is a differential equation of the form: dy Py Qy n dx This is solved by: (a) Divide both sides by yn to give: dy yn Py1 n Q dx (b) Let z = y1−n so that: dz dy (1 n) y n dx dx STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Bernoulli’s equation Substitution yields: dz dy (1 n) y n dx dx then: dy (1 n) y n Py1 n (1 n)Q dx becomes: dz P z Q1 1 dx Which can be solved using the integrating factor method. STROUD Worked examples and exercises are in the text Programme 24: First-order differential equations Learning outcomes Recognize the order of a differential equation Appreciate that a differential equation of order n can be derived from a function containing n arbitrary constants Solve certain first-order differential equations by direct integration Solve certain first-order differential equations by separating the variables Solve certain first-order homogeneous differential equations by an appropriate substitution Solve certain first-order differential equations by using an integrating factor Solve Bernoulli’s equation. STROUD Worked examples and exercises are in the text