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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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 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)}

Àwọn ìṣàmúlò-ètò

\frac{d}{d x} y{\left(x \right)} = 0.05 y{\left(x \right)}
The differential equation.
\text{order } 1
Order 1: the highest derivative present.
\text{Separable}
Move everything in y to one side with dy and everything in x to the other, then integrate both sides.
1.0 r - 0.05 = 0
Characteristic equation of the homogeneous part (substitute y = e^{rx}).
r = 0.05
Its roots.
y{\left(x \right)} = C_{1} e^{0.05 x}
General solution (C₁, C₂ … are arbitrary constants).
\checkmark
Verified: substituting the solution back into the equation gives 0.

Fi àwọn àgbèwọlé hàn

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

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Bii o ṣe le: Modelling with differential equations

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

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú Differential Equations

Separable equations First-order linear equations Second-order, constant coefficients Nonhomogeneous equations