NVCC MTH 267 Exam 3 Review

Focus is on formulas

u^’₁

$\frac{-y_2g\left(x\right)}{W}_{}$ (Variation of Parameters)

u^’₂

$\frac{y_1g\left(x\right)}{W}$ (Variation of Parameters)

Reduction of Order Formula

y_2=y_1\left(x\right)\int_{}^{}\frac{e^{-\int_{}^{}P\left(x\right)\differentialD x}}{y_1^2\left(x\right)}\differentialD x (To find 2nd sol. in the Fun. Sol. Set of 2nd Order Linear ODEs)

cosh(x) Identity

$$\frac{e^{x}+e^{-x}}{2}$$

sinh(x) Identity

$$\frac{e^{x}-e^{-x}}{2}$$

Derivative and Integral of sinh(x)

Always cosh(x)

Derivative and Integral of cosh(x)

Always sinh(x)

Linear ODE Aux Eq (y_c) Rational Sol. Format

$y=c_1e^{mx}+c_2xe^{mx}$, where m is the sol. to the aux eq

Linear ODE Aux Eq (y_c) Irrational Sol. Format

$$y=c_1e^{\alpha x}\cos\left(\beta x\right)+c_2e^{\alpha x}\sin\left(\beta x\right),m=\alpha\pm\beta i$$

Variation of Parameters Req.

Must be in standard form (leading coeff. = 1)

$\sin^2\left(x\right)$ power reduction identity

$$\frac{1-\cos\left(2x\right)}{2}$$

$\cos^2\left(x\right)$ power reduction identity

$$\frac{1+\cos\left(2x\right)}{2}$$

SHM Constants Formulas

$$\omega^2=\frac{k}{m};k=\frac{W}{s};2\lambda=\frac{\beta}{m};W=mg;P=\frac{2\pi}{\omega}\left(\frac{\sec}{cycle}\right);F=\frac{\omega}{2\pi}\left(\frac{cycles}{\sec}\right)$$

Dampened SHM Case 1

\lambda^2-\omega^2>0;x\left(t\right)=e^{-\lambda t}\left(c_1e^{\sqrt{\left(\lambda^2-\omega^2\right)}t}+c_2e^{-\sqrt{\lambda^2-\omega_{}^2}t}\right)

Free Damped Motion Case 2

$$\lambda^2-\omega^2=0;x\left(t\right)=e^{-\lambda t}\left(c_1+c_2t\right)$$

Free Damped Motion Case 3

\lambda^2-\omega^2<0;x\left(t\right)=e^{-\lambda t}\left(c_1\cos\left(\sqrt{\omega^2-\lambda^2}t\right)+c_2\sin\left(\sqrt{\omega^2-\lambda^2}t\right)\right)

Free Dampened Motion General ODE

x(t) = mx’’ + betax(x’) +kx = 0

Variation of Parameters y_p Formula

$$y_{p}=u_1\left(x\right)y_1\left(x\right)+u_2\left(x\right)y_2\left(x\right)$$

Wronksian

Solution set is linearly dependant if W = 0

Separable Differential Equation

A differential equation that can be expressed as $\frac{dy}{dx} = g(y)h(x)$, allowing integration of both sides.

Linear ODE Standard Form

Expressed as $\frac{dy}{dx} + P(x)y = Q(x)$ , where $P(x)$ and $Q(x)$ are functions of $x$. Solve using integrating factor:u\left(x\right)=e^{\int_{}^{}P\left(x\right)\differentialD x} → \frac{d}{\differentialD x}\left(u\left(x\right)y\right)=u\left(x\right)Q\left(x\right)

Homogeneous ODE

A differential equation of the form \frac{dy}{\differentialD x}=F\left(\frac{y}{x}\right) or \frac{\differentialD x}{\differentialD y}=F\left(\frac{x}{y}\right) that can be solved by substituting $u=\frac{y}{x}$ or $u=\frac{x}{y}$ . To identify, convert to $M(x,y)dx + N(x,y)dy = 0$ and ensure all terms have same total degree

Bernoulli ODE

Form given by $y' + P(x)y = Q(x)y^n$; solved using substitution to reduce it to a linear ODE. To solve, divide by yⁿ, then make the substitution u = y^1-n; u’ = (1-n)^-ny^-ny’

Exact Differential Equation

An equation of the form $M(x,y)dx + N(x,y)dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Exactifying an ODE

Use an integrating factor: u\left(x\right)=e^{\int_{}^{}\left(\frac{M_{y}-N_{x}}{N}\right)\differentialD x} OR u\left(y\right)=e^{\int_{}^{}\left(\frac{N_{x}-M_{Y}}{M}\right)\differentialD y}

Euler’s Method Formula

An approximation for solutions of ODEs, given by $y_{n+1} = y_n + hf(x_n, y_n)$, where $h$ is the step size.

Work Formula in Physics

Given by $W = Fd\cos(\theta)$ OR W=\int_{a}^{b}F\left(x\right)\differentialD x; represents work done by a force $F$ over distance $d$ at angle $\theta$.

Tangent Plane Equation

Describes the plane tangent to the surface at point $P(a,b,c)$, given by $z - c = f_x(a,b)(x-a) + f_y(a,b)(y-b)$.

Total Differential Equation

Indicates how a function changes in response to changes in its variables, given by $dz = f_x dx + f_y dy$.

Arc Length Formula

The length of a curve from $x=a$ to $x=b$ is given by $L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$.

Surface Area of Revolution

For surface around the x-axis: $S = 2\pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx$; similar for y-axis.

Radioactive Decay Formula

Describes the decay of a substance: $A = A_0 e^{kt}$, where $A_0$ is the initial amount and $k$ is a decay constant.

Newton’s Law of Cooling

Describes the rate of temperature change: $\frac{dT}{dt} = k(T - M)$, where $M$ is the ambient temperature.

Salt/Water Mixture Change

Model of concentration change: $\frac{dA}{dt} = R_{in} - R_{out}$ with $R_{in}$ and $R_{out}$ defined by their respective flow rates. $R_{IN}=Conc._{IN}\cdot FR_{IN}$ and $R_{OUT}=\frac{A}{InitialVol\pm NetRate}\cdot FR_{OUT}$

Kirchhoff’s Voltage Law

Total supplied voltage = total voltage in circuit: $E\left(t\right)=\frac{q}{c}+Rq^{\prime}+Lq$’’ , where $V_{CAP}=\frac{q}{c};current:i=q^{\prime}\left(t\right);V_{RES}=Vi$

\int_{}^{}\ln\left(x\right)\differentialD x

$$x\ln\left(x\right)-x+c$$

Product to Sum Identity for sin(A)cos(B)

$$\frac{1}{2} [\text{sin}(A+B) + \text{sin}(A-B)]$$

Product to Sum Identity for cos(A)cos(B)

$$\frac{1}{2}[\text{cos}(A-B)+\cos\left(A+B\right)]$$

Product to Sum Identity for sin(A)sin(B)

$$\frac{1}{2} [\text{cos}(A-B) - \text{cos}(A+B)]$$