Maclaurin Series from Differential Equations (DP IB Maths: AA HL)

Can I apply Maclaurin Series to solving differential equations?

• If you have a differential equation of the form  along with the value of it is possible to build up the Maclaurin series of the solution term by term
• This does not necessarily tell you the explicit function of  that corresponds to the Maclaurin series you are finding
• But the Maclaurin series you find is the exact Maclaurin series for the solution to the differential equation
• The Maclaurin series can be used to approximate the value of the solution y = f(x) for different values of
• You can increase the accuracy of this approximation by calculating additional terms of the Maclaurin series for higher powers of

How can I find the Maclaurin Series for the solution to a differential equation?

• STEP 1: Use implicit differentiation to find expressions for  etc., in terms of and lower-order derivatives of  
  • The number of derivatives you need to find depends on how many terms of the Maclaurin series you want to find
  • For example, if you want the Maclaurin series up to the  term, then you will need to find derivatives up to  (the fourth derivative of )
• STEP 2: Using the given initial value for , find the values of etc., one by one 
  • Each value you find will then allow you to find the value for the next higher derivative
• STEP 3: Put the values found in STEP 2 into the general Maclaurin series formula

• • This formula is in your exam formula booklet
  • is the solution to the differential equation, so  corresponds to in the formula, corresponds to , and so on
• STEP 4: Simplify the coefficients for each of the powers of in the resultant Maclaurin series