Differential equations flashcards

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What is the general form of a first-order linear differential equation?

$\frac{dy}{dx} + P(x)y = Q(x)$, where $P(x)$ and $Q(x)$ are functions of $x$.

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What is an Integrating Factor (IF)?

A function, $I(x)$, which is multiplied through a first-order linear DE to make the left-hand side the exact derivative of a product. It is calculated as $I(x) = e^{\int P(x) dx}$.

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After multiplying by the IF, $I(x)$, what does the LHS of $\frac{dy}{dx} + P(x)y = Q(x)$ become?

The derivative of the product of the integrating factor and $y$. That is, $\frac{d}{dx}(I(x)y)$.

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What is the Auxiliary Equation for $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = 0$?

A quadratic equation $am^2 + bm + c = 0$, whose roots determine the form of the complementary function.

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What is the Complementary Function (CF) if the auxiliary equation has two distinct real roots, $m_1$ and $m_2$?

$y_{CF} = Ae^{m_1x} + Be^{m_2x}$

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What is the CF if the auxiliary equation has one repeated real root, $m$?

$y_{CF} = (Ax+B)e^{mx}$

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What is the CF if the auxiliary equation has complex conjugate roots, $p \pm iq$?

$y_{CF} = e^{px}(A\cos(qx) + B\sin(qx))$

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What is the General Solution of a non-homogeneous DE?

General Solution = Complementary Function (CF) + Particular Integral (PI). $y = y_{CF} + y_{PI}$.

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What is the trial form for the Particular Integral (PI) if the right-hand side is $ke^{ax}$?

Try $y_{PI} = Ce^{ax}$. A common trap is if $e^{ax}$ is already a term in the CF. In that case, you must try $y_{PI} = Cxe^{ax}$ (or $Cx^2e^{ax}$ if the CF root was repeated).

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What is the trial form for the PI if the right-hand side is a polynomial of degree $n$, e.g., $3x^2+5$?

A general polynomial of the same degree. For $3x^2+5$, try $y_{PI} = Cx^2 + Dx + E$.

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What is the trial form for the PI if the right-hand side is $k\sin(ax)$ or $k\cos(ax)$?

A combination of both sine and cosine: $y_{PI} = C\cos(ax) + D\sin(ax)$.