Differential Equations and Integral Transforms Flashcards

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Differential Equations and Integral Transforms

Ordinary differential equations (ODEs), including first-order types, second-order linear equations, Laplace transforms, and systems of equations.

What is the order of a differential equation

The highest derivative present in the equation

What defines a linear differential equation

The dependent variable and its derivatives appear only to the first power and are not multiplied together

What is the integrating factor for $y^{'} + P (x) y = Q (x)$

$μ (x) = e^{\int P (x) d x}$

What is an autonomous differential equation

An equation where the independent variable does not appear explicitly, written as $\frac{d y}{d t} = f (y)$

What is the characteristic equation for $a y^{''} + b y^{'} + cy = 0$

$a r^{2} + b r + c = 0$

What is the general solution if the characteristic equation has two distinct real roots $r_{1}$ and $r_{2}$

$y = c_{1} e^{r_{1} x} + c_{2} e^{r_{2} x}$

What is the general solution if the characteristic equation has a repeated real root $r$

$y = (c_{1} + c_{2} x) e^{r x}$

What is the general solution if the characteristic equation has complex roots $α \pm i β$

$y = e^{αx} (c_{1} cos (β x) + c_{2} sin (β x))$

What is the Wronskian of two functions $y_{1}$ and $y_{2}$

$W (y_{1}, y_{2}) = y_{1} y_{2}^{'} - y_{2} y_{1}^{'}$

What does a non zero Wronskian imply for a set of solutions

The solutions are linearly independent

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