Differential Equations - First-Order & Second-Order Equations 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. Example: $\frac{dy}{dx} = xy$.

Question 2

How is the order of a differential equation determined?

Accepted Answer

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

Question 3

Distinguish between a linear and a non-linear ODE.

Accepted Answer

A linear ODE can be written in the form $a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \dots + a_1(x)y' + a_0(x)y = g(x)$. The dependent variable $y$ and its derivatives appear only to the first power and are not multiplied together. Any other form is non-linear. Example linear: $y'' + xy' + y = \sin(x)$. Example non-linear: $y'' + (y')^2 = 0$.

Question 4

Explain the difference between a homogeneous and a non-homogeneous ODE.

Accepted Answer

For a linear ODE, if $g(x) = 0$, the equation is homogeneous. If $g(x) 
eq 0$, it is non-homogeneous. For example, $y'' + 2y' + y = 0$ is homogeneous, while $y'' + 2y' + y = e^x$ is non-homogeneous. (Note: For first-order ODEs, 'homogeneous' can also refer to $dy/dx = f(y/x)$.)

Question 5

What is an Initial Value Problem (IVP)?

Accepted Answer

An ODE coupled with a set of conditions that specify the value of the unknown function and its derivatives at a single point. For a first-order ODE $y' = f(x,y)$, an IVP includes $y(x_0) = y_0$. For a second-order ODE, it includes $y(x_0) = y_0$ and $y'(x_0) = y_1$.

Question 6

What is a Boundary Value Problem (BVP)?

Accepted Answer

An ODE coupled with a set of conditions that specify the value of the unknown function or its derivatives at two or more different points (boundaries). For a second-order ODE, a BVP might include $y(x_0) = y_0$ and $y(x_1) = y_1$.

Question 7

Describe the method of Separation of Variables for first-order ODEs.

Accepted Answer

This method applies to ODEs that can be written in the form $\frac{dy}{dx} = f(x)g(y)$. The variables are separated to get $\frac{dy}{g(y)} = f(x)dx$, and then both sides are integrated.

Question 8

How do you solve a first-order linear ODE using an Integrating Factor?

Accepted Answer

For a first-order linear ODE in the standard form $\frac{dy}{dx} + P(x)y = Q(x)$, the integrating factor is $\mu(x) = e^{\int P(x)dx}$. Multiplying the entire equation by $\mu(x)$ makes the left side the derivative of a product: $\frac{d}{dx}[\mu(x)y] = \mu(x)Q(x)$. Integrate both sides to solve for $y$.

Question 9

What is an exact differential equation, and how is it solved?

Accepted Answer

A first-order ODE $M(x,y)dx + N(x,y)dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$. The solution is given by $F(x,y) = C$, where $\frac{\partial F}{\partial x} = M(x,y)$ and $\frac{\partial F}{\partial y} = N(x,y)$.

Question 10

How can you test if a first-order ODE $M(x,y)dx + N(x,y)dy = 0$ is exact?

Accepted Answer

An ODE is exact if and only if the partial derivative of $M(x,y)$ with respect to $y$ equals the partial derivative of $N(x,y)$ with respect to $x$. That is, $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.

Question 11

What is the general form of a second-order linear homogeneous ODE with constant coefficients?

Accepted Answer

It is an equation of the form $ay'' + by' + cy = 0$, where $a, b, c$ are real constants and $a 
eq 0$.

Question 12

How do you find the general solution for $ay'' + by' + cy = 0$ when the characteristic equation has distinct real roots?

Accepted Answer

The characteristic equation is $ar^2 + br + c = 0$. If it has distinct real roots $r_1$ and $r_2$, the general solution is $y(x) = c_1e^{r_1x} + c_2e^{r_2x}$.

Question 13

How do you find the general solution for $ay'' + by' + cy = 0$ when the characteristic equation has repeated real roots?

Accepted Answer

If the characteristic equation $ar^2 + br + c = 0$ has a repeated real root $r_1$ (i.e., $r_1 = r_2 = -b/(2a)$), the general solution is $y(x) = c_1e^{r_1x} + c_2xe^{r_1x}$.

Question 14

How do you find the general solution for $ay'' + by' + cy = 0$ when the characteristic equation has complex conjugate roots?

Accepted Answer

If the characteristic equation $ar^2 + br + c = 0$ has complex conjugate roots $r = \alpha \pm i\beta$, the general solution is $y(x) = e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))$.

Question 15

Explain the general approach of the Method of Undetermined Coefficients for non-homogeneous second-order ODEs.

Accepted Answer

For $ay'' + by' + cy = g(x)$, where $g(x)$ is a polynomial, exponential, sine, or cosine (or a product/sum of these), we guess a particular solution $y_p$ of the same form as $g(x)$ (with undetermined coefficients). Substitute $y_p$ into the ODE to find the coefficients. The general solution is $y = y_c + y_p$, where $y_c$ is the complementary solution from the homogeneous equation.

Question 16

Give examples of trial solutions $y_p$ for the Method of Undetermined Coefficients for different $g(x)$ forms.

Accepted Answer

If $g(x) = P_n(x)$ (polynomial of degree $n$), try $y_p = A_nx^n + \dots + A_0$. If $g(x) = Ce^{kx}$, try $y_p = Ae^{kx}$. If $g(x) = C\cos(kx)$ or $C\sin(kx)$, try $y_p = A\cos(kx) + B\sin(kx)$. (Modification rule: If any term in $y_p$ is part of $y_c$, multiply $y_p$ by $x$ or $x^2$.)

Question 17

When is the Method of Variation of Parameters preferred over Undetermined Coefficients?

Accepted Answer

Variation of Parameters is a more general method applicable to any linear non-homogeneous ODE, including those with variable coefficients or when $g(x)$ is not of the specific forms required for Undetermined Coefficients (e.g., $g(x) = 	an(x)$ or $g(x) = \frac{1}{x}$).

Question 18

State the formula for a particular solution $y_p$ using the Method of Variation of Parameters for $y'' + P(x)y' + Q(x)y = g(x)$.

Accepted Answer

If $y_c = c_1y_1(x) + c_2y_2(x)$ is the complementary solution, then $y_p = -y_1(x)\int \frac{y_2(x)g(x)}{W(y_1, y_2)}dx + y_2(x)\int \frac{y_1(x)g(x)}{W(y_1, y_2)}dx$, where $W(y_1, y_2) = y_1y_2' - y_2y_1'$ is the Wronskian.

Question 19

What is a Bernoulli equation, and how is it solved?

Accepted Answer

A Bernoulli equation is a first-order non-linear ODE of the form $\frac{dy}{dx} + P(x)y = Q(x)y^n$, where $n$ is any real number. It can be transformed into a linear ODE by substituting $u = y^{1-n}$, which yields $\frac{du}{dx} + (1-n)P(x)u = (1-n)Q(x)$.

Question 20

What is a homogeneous first-order ODE (in the context of $dy/dx = f(x,y)$), and how is it solved?

Accepted Answer

A first-order ODE is homogeneous if $f(tx,ty) = f(x,y)$ for some $t$, meaning $f(x,y)$ can be expressed as a function of $y/x$. It is solved by the substitution $v = y/x$ (so $y = vx$ and $dy/dx = v + x dv/dx$), which transforms it into a separable equation in terms of $v$ and $x$.

Question 21

Briefly describe the Existence and Uniqueness Theorem for first-order IVPs.

Accepted Answer

For the IVP $y' = f(x,y)$, $y(x_0) = y_0$, if $f(x,y)$ and $\frac{\partial f}{\partial y}$ are continuous in some rectangle $R$ containing $(x_0, y_0)$, then there exists a unique solution $y(x)$ on some interval $(x_0-h, x_0+h)$ within $R$.

Differential Equations - First-Order & Second-Order Equations

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