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Achievement objective M8-12

In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:

  • form differential equations and interpret the solutions.


  • Uses knowledge of anti-differentiation to form and solve differential equations of the type:
    • with the rate of change directly or inversely proportional to the variable of interest
    • first and second order with variables easily separated for functions including those listed in M8-11.
  • Applies boundary or initial conditions to solutions of differential equations.
  • Distinguishes between families of solutions and exact solutions using given boundary or initial conditions and interprets these solutions.
  • Links families of solutions to the gradient field of the differential equation.
  • See key mathematical ideas on NZmaths.


M8-12 links from M7-10.

Possible context elaborations

  •   dy by dx equals y times sin x.
  •   d squared y by d x squared equals “polynomial”
  •   dy by dx equals e to the power of x.
  •   dy by dx equals one divided by x.

Situations which students will be expected to form a model for, and solve, include:

Assessment for qualifications

NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the NZQA website.

The NZQA subject-specific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.

Learn more:

The following achievement standard(s) could assess learning outcomes from this AO:

  • AS91579 Mathematics and statistics 3.7 Apply integration methods in solving problems; External, 6 credits.

Refer to the mathematics and statistics matrix.

Last updated September 26, 2013