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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.

ຕົວຢ່າງທີ່ໄດ້ເຮັດ · y' + y = x

Differential-Equations · Differential-Equations

Y' + y = x

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

ຂັ້ນຕອນຕໍ່ຂັ້ນຕອນ

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.

ເປີດເຜີຍຄຳຕອບ

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

ປັດຈຸບັນທ່ານ ເລືອກບັນຫາ ຫຼື ພິມ ຫຼື ແຕ້ມເອງ. ທຸກບາດກ້າວ, ຮູບ, ຄໍາຕອບຖືກລົບຈົນກວ່າເຈົ້າຈະຖາມມັນ.

ຮັກສາການເຮັດວຽກຂອງທ່ານໄວ້

ບັນຊີຟຣີ ເພີ່ມຂໍ້ມູນໃສ່ບົດຮຽນທຸກບົດ, ບັນທຶກຂອງສິ່ງທີ່ທ່ານໄດ້ເຮັດແລ້ວ, ບັນຫາທີ່ທ່ານໄດ້ແກ້ໄຂໄວ້ໃນບ່ອນດຽວ ແລະ ຄູສອນທີ່ທ່ານສາມາດຖາມກ່ຽວກັບໜ້ານີ້. ຄະນິດສາດເອງແມ່ນເປີດໃຫ້ທຸກຄົນ, ບໍ່ວ່າຈະໄດ້ເຂົ້າສູ່ລະບົບ ຫຼື ບໍ່.

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ຕົວສະແດງທີ່ໃຊ້ຢູ່ທີ່ນີ້

ກົດຕົວອັກສອນໃດໜຶ່ງເພື່ອເບິ່ງຄວາມໝາຍເຕັມ, ຮູບ ແລະ ຕົວອັກສອນແຕ່ລະຕົວໃນມັນໝາຍຄວາມວ່າແນວໃດ.

ວິທີການ: 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.

ຄໍາຖາມທີ່ຄົນຖາມ

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.

ເພີ່ມເຕີມໃນ Differential Equations

Separable equations Second-order, constant coefficients Nonhomogeneous equations Modelling with differential equations