Using substitution to reduce a differential equation to a known form

Using substitution to reduce a differential equation to a known form

Common Question Formats

  • Given a differential equation that is complex in appearance, use substitution to transform it into a standard or known form that’s easier to solve.
  • Apply a specific substitution to a given differential equation to facilitate integration or differentiation.
  • Solve a first order homogeneous differential equation by using a substitution that will turn it into a separable equation.
  • Given a differential equation in terms of x and y, use an appropriate substitution to reduce the equation to a function of a single variable, which simplifies the process of solving.

Strategies for Addressing Questions

  • When faced with complex differential equations, it is important to identify whether it can be simplified into a standard form using substitution. Examples of these forms include the separable, linear or homogeneous differential equations.
  • In some cases, the substitution needed to simplify the equation won’t be obvious. Try using common substitutions like v = y/x, v = y’+p, u = ax+by+c, or y = vu for a first order homogeneous differential equation to see if they result in simplification.
  • Once a substitution is made, always ensure to replace every instance of the original variable in the equation. Don’t forget to also change the derivative accordingly.
  • After the substitution process, the resultant equation should be simple enough to – integrate, differentiate or otherwise – solve using standard methods.

Potential Pitfalls

  • It’s easy to forget to change the derivative when making a substitution. Remember that if for example you replace y with vx, then dy/dx is not just dv/dx but (v + x dv/dx).
  • Make sure to correctly write all steps of substitution. Failing to do so might lead to errors in the final answer.
  • After solving for the new substitute variable, remember to substituting back to get the solution in terms of original variables. Not doing so, can result in losing marks.
  • Ensure to check your solutions by substituting back into the original differential equation.