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Modelling with differential equations

Growth, decay, cooling, mixing: writing the equation from the sentence.

Every model starts with a sentence of the form “the rate of change of y is proportional to …”. Growth at 5% a year is y′ = 0.05y; cooling is T′ = −k(T − T_room); a mixing tank balances what flows in against what flows out. Write the sentence as an equation, solve, then fit the constant to a known value.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · y' = 0.05*y

Differential-Equations · Differential-Equations

Y' = 0.05*y

\frac{d}{d x} y{\left(x \right)} = 0.05 y{\left(x \right)}

ជំហាន​ដោយ​ជំហាន

  1. \frac{d}{d x} y{\left(x \right)} = 0.05 y{\left(x \right)}

    The differential equation.

  2. \text{order } 1

    Order 1: the highest derivative present.

  3. \text{Separable}

    Move everything in y to one side with dy and everything in x to the other, then integrate both sides.

  4. 1.0 r - 0.05 = 0

    Characteristic equation of the homogeneous part (substitute y = e^{rx}).

  5. r = 0.05

    Its roots.

  6. y{\left(x \right)} = C_{1} e^{0.05 x}

    General solution (C₁, C₂ … are arbitrary constants).

  7. \checkmark

    Verified: substituting the solution back into the equation gives 0.

បង្ហាញ​ចម្លើយ
y{\left(x \right)} = C_{1} e^{0.05 x}

ឥឡូវ​អ្នក ជ្រើស​បញ្ហា ឬ​វាយ​បញ្ចូល ឬ​គូរ​របស់​អ្នក​ផ្ទាល់ ។ ជំហាន​នីមួយៗ រូបភាព ចម្លើយ​ត្រូវ​បាន​លាក់​រហូត​ដល់​អ្នក​សួរ ។

រក្សា​ការ​ធ្វើការ​របស់​អ្នក​ផ្ទាល់

គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

ចុះឈ្មោះ ចូល

និមិត្ត​សញ្ញា​ដែល​បាន​ប្រើ​នៅ​ទីនេះ

ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: Modelling with differential equations

  1. The differential equation.
  2. Order 1: the highest derivative present.
  3. Move everything in y to one side with dy and everything in x to the other, then integrate both sides.
  4. Characteristic equation of the homogeneous part (substitute y = e^{rx}).
  5. Its roots.
  6. General solution (C₁, C₂ … are arbitrary constants).
  7. Verified: substituting the solution back into the equation gives 0.

សំណួរ​ដែល​មនុស្ស​សួរ

What is a differential equation?

An equation whose unknown is a function, relating it to its own derivatives. "The rate of growth is proportional to the population" is y′ = ky, and solving it means finding y as a function of time.

Why does the solution have arbitrary constants?

Integrating loses information: many functions share the same derivative. An n-th order equation has n constants, fixed by n initial or boundary conditions.

បន្ថែម​ទៀត​ក្នុង Differential Equations

Separable equationsFirst-order linear equationsSecond-order, constant coefficientsNonhomogeneous equations