Differential Equations - First & Second Order Solutions Flashcards

Question 1

What is an Ordinary Differential Equation (ODE)?

Accepted Answer

An equation involving an unknown function of a single independent variable and its derivatives with respect to that variable. For example, $y' + 2y = \sin(x)$.

Question 2

What is the 'order' of a differential equation?

Accepted Answer

The order of the highest derivative present in the equation. For example, $y'' + 3y' - y = x^2$ is a second-order ODE.

Question 3

When is a differential equation considered 'linear'?

Accepted Answer

A differential equation is linear if the dependent variable and all its derivatives appear only to the first power and are not multiplied together. The coefficients can be functions of the independent variable. A general linear $n$-th order ODE is $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \dots + a_1(x)y' + a_0(x)y = g(x)$.

Question 4

Describe the method of 'Separable Variables' for first-order ODEs.

Accepted Answer

A first-order ODE is separable if it can be written in the form $g(y)dy = f(x)dx$. The solution is found by integrating both sides: $\int g(y)dy = \int f(x)dx + C$.

Question 5

What is an 'Exact Differential Equation'?

Accepted Answer

A first-order ODE of the form $M(x,y)dx + N(x,y)dy = 0$ is exact if there exists a function $F(x,y)$ such that $dF = Mdx + Ndy$. This implies that $M = rac{\partial F}{\partial x}$ and $N = rac{\partial F}{\partial y}$.

Question 6

How do you test for exactness of a first-order ODE $M(x,y)dx + N(x,y)dy = 0$?

Accepted Answer

An ODE is exact if and only if $rac{\partial M}{\partial y} = rac{\partial N}{\partial x}$.

Question 7

How do you solve an 'Exact Differential Equation'?

Accepted Answer

1. Verify exactness: $rac{\partial M}{\partial y} = rac{\partial N}{\partial x}$.
2. Integrate $M$ with respect to $x$ (treating $y$ as constant) to find $F(x,y) = \int M(x,y)dx + g(y)$.
3. Differentiate $F(x,y)$ with respect to $y$ and set it equal to $N(x,y)$ to find $g'(y)$.
4. Integrate $g'(y)$ to find $g(y)$.
5. The general solution is $F(x,y) = C$.

Question 8

What is a 'Linear First-Order ODE' and how is an 'Integrating Factor' used?

Accepted Answer

A linear first-order ODE has the form $y' + P(x)y = Q(x)$. An integrating factor $\mu(x)$ is multiplied to both sides to make the left side the derivative of a product, allowing direct integration. The solution is $y = rac{1}{\mu(x)}\int \mu(x)Q(x)dx$.

Question 9

How do you find the integrating factor $\mu(x)$ for $y' + P(x)y = Q(x)$?

Accepted Answer

The integrating factor is given by $\mu(x) = e^{\int P(x)dx}$.

Question 10

What is an Initial Value Problem (IVP) for a first-order ODE?

Accepted Answer

An IVP consists of a first-order ODE along with an initial condition, typically $y(x_0) = y_0$. This condition is used to find the unique particular solution by determining the constant of integration.

Question 11

What is the general form of a Second-Order Linear Homogeneous ODE with constant coefficients?

Accepted Answer

The general form is $ay'' + by' + cy = 0$, where $a, b, c$ are real constants and $a 
eq 0$.

Question 12

What is the 'Characteristic Equation' for $ay'' + by' + cy = 0$?

Accepted Answer

The characteristic equation (or auxiliary equation) is $ar^2 + br + c = 0$. Its roots determine the form of the general solution.

Question 13

How do you find the general solution when the characteristic equation has 'distinct real roots' $r_1, r_2$?

Accepted Answer

The general solution is $y(x) = C_1e^{r_1x} + C_2e^{r_2x}$.

Question 14

How do you find the general solution when the characteristic equation has 'repeated real roots' $r_1 = r_2 = r$?

Accepted Answer

The general solution is $y(x) = C_1e^{rx} + C_2xe^{rx}$.

Question 15

How do you find the general solution when the characteristic equation has 'complex conjugate roots' $\alpha \pm ieta$?

Accepted Answer

The general solution is $y(x) = e^{\alpha x}(C_1\cos(eta x) + C_2\sin(eta x))$.

Question 16

What is the general form of a Second-Order Linear Non-Homogeneous ODE?

Accepted Answer

The general form is $ay'' + by' + cy = g(x)$, where $g(x) 
eq 0$. The general solution is $y(x) = y_c(x) + y_p(x)$, where $y_c$ is the complementary solution (solution to the homogeneous part) and $y_p$ is a particular solution.

Question 17

Explain the 'Method of Undetermined Coefficients' for $ay'' + by' + cy = g(x)$.

Accepted Answer

This method finds a particular solution $y_p(x)$ by making an educated guess based on the form of $g(x)$. The guess contains unknown coefficients, which are determined by substituting $y_p$ and its derivatives into the ODE and equating coefficients.

Question 18

What are common initial guesses for $Y_p(x)$ in Undetermined Coefficients when $g(x)$ is a polynomial?

Accepted Answer

If $g(x)$ is a polynomial of degree $n$, say $P_n(x)$, the initial guess for $Y_p(x)$ is a general polynomial of the same degree: $A_nx^n + A_{n-1}x^{n-1} + \dots + A_0$.

Question 19

What are common initial guesses for $Y_p(x)$ in Undetermined Coefficients when $g(x)$ is an exponential function?

Accepted Answer

If $g(x) = ke^{\alpha x}$, the initial guess for $Y_p(x)$ is $Ae^{\alpha x}$.

Question 20

What are common initial guesses for $Y_p(x)$ in Undetermined Coefficients when $g(x)$ is a sine/cosine function?

Accepted Answer

If $g(x) = k_1\cos(eta x) + k_2\sin(eta x)$, the initial guess for $Y_p(x)$ is $A\cos(eta x) + B\sin(eta x)$.

Question 21

When do you need to modify the initial guess for $Y_p(x)$ in Undetermined Coefficients?

Accepted Answer

If any term in the initial guess for $Y_p(x)$ is already part of the complementary solution $y_c(x)$, the guess must be multiplied by $x$ (or $x^2$ if it's a repeated root) until no term in $Y_p(x)$ is a solution to the homogeneous equation.

Question 22

Briefly explain the 'Method of Variation of Parameters' for $y'' + p(x)y' + q(x)y = f(x)$.

Accepted Answer

This method finds a particular solution $y_p(x)$ for non-homogeneous linear ODEs, especially when $g(x)$ is not suitable for undetermined coefficients. It assumes $y_p(x) = u_1(x)y_1(x) + u_2(x)y_2(x)$, where $y_1, y_2$ are the fundamental solutions of the homogeneous equation, and $u_1, u_2$ are functions to be determined by specific formulas involving integrals.

Question 23

When is 'Variation of Parameters' preferred over 'Undetermined Coefficients'?

Accepted Answer

Variation of Parameters is more general and can be used when the coefficients $p(x), q(x)$ are not constant, or when $g(x)$ is not a simple polynomial, exponential, or trigonometric function (or combinations thereof), making Undetermined Coefficients impractical or impossible.

Question 24

What is the 'Superposition Principle' for non-homogeneous linear ODEs?

Accepted Answer

If $g(x)$ is a sum of functions, $g(x) = g_1(x) + g_2(x) + \dots + g_k(x)$, then the particular solution $y_p(x)$ can be found as the sum of particular solutions for each $g_i(x)$ individually: $y_p(x) = y_{p1}(x) + y_{p2}(x) + \dots + y_{pk}(x)$.

Question 25

How do you solve an Initial Value Problem (IVP) for a second-order ODE?

Accepted Answer

1. Find the general solution $y(x) = y_c(x) + y_p(x)$, which will contain two arbitrary constants $C_1, C_2$.
2. Use the two initial conditions, typically $y(x_0) = y_0$ and $y'(x_0) = y_1$, to form a system of two linear equations for $C_1$ and $C_2$.
3. Solve for $C_1$ and $C_2$ and substitute them back into the general solution to get the unique particular solution.

Differential Equations - First & Second Order Solutions

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