1 First Order Linear Differential Equations Chapter 4 FP2 EdExcel

2 Splitting the Variables This is a method used in C4 of the Core Mathematics A level and then in Chapter 4 FP2. When: We can split the variable in order to find a final equation in the form : y=f(x) We do this by Dividing and Multiplying to get g(y) on one side and f(x) on the other side, and splitting dy/dx up to the appropriate sides like so: We can now integrate to find y in the form f(x). Do not forget the arbitrary constant when doing so, and if given values for x and y, use them to find the value of this constant. Worked example (Elmwood C4, paper A)

3 dy/dx +p(x)y=q(x) Multiplying Factors The problem in such a differential equation is that one can not simply divide and multiply to split the variables. We therefore need to manipulate the expression to allow calculus to be used to solve the problem. In order to do this we need to think back to C3 calculus and remember the product rule. We need to manipulate the equation into this form, and we can do this by using a multiplying factor, M(x). By doing this, we have allowed ourselves to manipulate the equation into the form of the product rule, because we can let M’(x)=M(x)q(x), and thus find M(x). We know that: This gives us a generalised form of our Multiplying factor M(x). We can ignore arbitrary constants in finding M(x) as we only need a solution, not a particular one, and they would only cancel in the equation anyway.

4 Using the Multiplying factors We know that: We also know that: We now get the form: The LHS takes the form of the product rule and can be changed to: So we get: Which can be solved using calculus. Worked example (EdExcel FP1 Jan ’08):

5 Substitutions One can use substitutions in order to transform differential equations into more easily solvable problems. These substitutions eliminate y from the equation and create a differential equation in terms of x and z (where z is the substitution). »This can now be solved by splitting the variables as shown on page 2. One can also use substitutions in order to eliminate x and y and create a simple calculus problem to solve. »This can now be solved by splitting the variables as shown on page 2.