Analytical Solutions to Differential Equations (DP IB Maths: AA HL)

What is separation of variables?

• Separation of variables can be used to solve certain types of first order differential equations
• Look out for equations of the form 
• i.e.  is a function of multiplied by a function of
• be careful – the ‘function of ’ may just be a constant!
• For example in , and
• If the equation is in that form you can use separation of variables to try to solve it
• If the equation is not in that form you will need to use another solution method

How do I solve a differential equation using separation of variables?

• STEP 1: Rearrange the equation into the form 
• STEP 2: Take the integral of both sides to change the equation into the form 

• • You can think of this step as ‘multiplying the across and integrating both sides’
    • Mathematically that’s not quite what is actually happening, but it will get you the right answer here!
• STEP 3: Work out the integrals on both sides of the equation to find the general solution to the differential equation
  • Don’t forget to include a constant of integration
    • Although there are two integrals, you only need to include one constant of integration
  • Look out for integrals that require you to use partial fractions to solve them
    • See ‘Integrating with Partial Fractions’ in 5.9 Advanced Integration
• STEP 4: Use any boundary or initial conditions in the question to work out the value of the integration constant
• STEP 5: If necessary, rearrange the solution into the form required by the question

What is a homogeneous first order differential equation?

• If a first order differential equation can be written in the form  then it is said to be homogeneous

How do I solve a homogeneous first order differential equation?

• These equations can be solved using the substitution 
• STEP 1: If necessary, rearrange the equation into the form 
• STEP 2: Replace all instances of  in your equation with v
• STEP 3: Use the product rule and implicit differentiation to replace in your equation with 
• This is because 
• STEP 4: Solve your new differential equation to find the solution in terms of v and x
• You may need to use other methods for differential equations, such as separation of variables, at this stage
• STEP 5: Substitute  into the solution from Step 4, in order to find the solution in terms of y and x

What else should I know about solving homogeneous first order differential equations?

• After finding the solution in terms of y and x you may be asked to do other things with the solution
• For example you may be asked to find the solution corresponding to certain initial or boundary conditions
• Or you may be asked to express your answer in a particular form, such as y = f(x)
• It is sometimes possible to solve differential equations that are not homogeneous by using the substitution 
• For such a situation in an exam question, you would be told explicitly to use the substitution
• You would not be expected to know that you could use the substitution in a case where the differential equation was not homogeneous

What is an integrating factor?

• An integrating factor can be used to solve a differential equation that can be written in the standard form 
• Be careful – the ‘functions of x’ p(x) and q(x) may just be constants!
• For example in , p(x) = 6 and q(x) = e^-2x
• While in ,   and q(x) = 12
• For an equation in standard form, the integrating factor is 

How do I use an integrating factor to solve a differential equation?

• STEP 1: If necessary, rearrange the differential equation into standard form
• STEP 2: Find the integrating factor
• Note that you don’t need to include a constant of integration here when you integrate  ∫p(x) dx
• STEP 3: Multiply both sides of the differential equation by the integrating factor

• This will turn the equation into an exact differential equation of the form 

• STEP 4: Integrate both sides of the equation with respect to x
• The left side will automatically integrate to 
• For the right side, integrate  using your usual techniques for integration
• Don’t forget to include a constant of integration
• Although there are two integrals, you only need to include one constant of integration
• STEP 5: Rearrange your solution to get it in the form y = f(x)

What else should I know about using an integrating factor to solve differential equations?

• After finding the general solution using the steps above you may be asked to do other things with the solution
• For example you may be asked to find the solution corresponding to certain initial or boundary conditions