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First-order linear equations

y′ + P(x)y = Q(x) and the integrating factor that makes it integrable.

Multiply y′ + P(x)y = Q(x) by μ = e^{∫P dx} and the left side collapses to (μy)′, one integration finishes it. This is the workhorse for mixing tanks, RC circuits and Newton's law of cooling.

Àwọn ààtò àwọn ìṣàmúlò-ètò · y' + y = x

Differential-Equations · Differential-Equations

Y' + y = x

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

Àwọn ìṣàmúlò-ètò

y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = x
The differential equation.
\text{order } 1
Order 1: the highest derivative present.
\text{Exact}
M dx + N dy = 0 with ∂M/∂y = ∂N/∂x: find the potential F with F_x = M, F_y = N; the solution is F = C.
r + 1 = 0
Characteristic equation of the homogeneous part (substitute y = e^{rx}).
r = -1
Its roots.
y{\left(x \right)} = C_{1} e^{- x} + x - 1
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^{- x} + x - 1

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Bii o ṣe le: First-order linear equations

The differential equation.
Order 1: the highest derivative present.
M dx + N dy = 0 with ∂M/∂y = ∂N/∂x: find the potential F with F_x = M, F_y = N; the solution is F = C.
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 Second-order, constant coefficients Nonhomogeneous equations Modelling with differential equations