maths.free › Differential Equations › Nonhomogeneous equations

Nonhomogeneous equations

Driving forces: homogeneous solution plus a particular solution.

With a forcing term on the right, the general solution is the homogeneous solution plus any one particular solution. Undetermined coefficients guesses the particular solution in the shape of the forcing (exponential, polynomial, sinusoid); variation of parameters handles the rest.

Àwọn ààtò àwọn ìṣàmúlò-ètò · y'' - 3y' + 2y = e^x

Differential-Equations · Differential-Equations

Y'' - 3y' + 2y = e^x

2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = e^{x}

Àwọn ìṣàmúlò-ètò

2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = e^{x}
The differential equation.
\text{order } 2
Order 2: the highest derivative present.
\text{Undetermined coefficients}
Solve the homogeneous part from the characteristic equation, then guess a particular solution shaped like the right-hand side.
r^{2} - 3 r + 2 = 0
Characteristic equation of the homogeneous part (substitute y = e^{rx}).
r = 2,\ r = 1
Its roots.
y{\left(x \right)} = \left(C_{1} + C_{2} e^{x} - x\right) e^{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)} = \left(C_{1} + C_{2} e^{x} - x\right) e^{x}

Nígbà yìí Yan àwọn àkóónú kan, tàbí tẹ̀ àtí kọ́ àwọn àkóónú rẹ̀. Àwọn ìṣàmúlò-ètò gbogbo, àwòrán, àwọn à

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Bii o ṣe le: Nonhomogeneous equations

The differential equation.
Order 2: the highest derivative present.
Solve the homogeneous part from the characteristic equation, then guess a particular solution shaped like the right-hand side.
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 Modelling with differential equations