Differential Equations First-Order Ordinary Differential Equations Syllabus

First-Order Ordinary Differential Equations syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for First-Order Ordinary Differential Equations in Differential Equations, not a summary of it.

1. 1

   Introduction to Differential Equations

   4 topics

   • Definitions and Terminology
   • Solutions of a Differential Equation
   • Initial-Value and Boundary-Value Problems
   • Differential Equations as Mathematical Models
2. 2

   Separable and Exact Equations

   3 topics

   • Separable Variables
   • Exact Equations
   • Integrating Factors
3. 3

   Linear and Special First-Order Equations

   4 topics

   • Linear First-Order Equations
   • Bernoulli Equation
   • Homogeneous Equations and Substitutions
   • Riccati Equation
4. 4

   Qualitative Methods for First-Order Equations

   3 topics

   • Direction Fields and Slope Fields
   • Autonomous Equations and Phase Lines
   • Picard Iteration and Successive Approximations
5. 5

   Modeling with First-Order Equations

   5 topics

   • Growth and Decay Models
   • Newton's Law of Cooling
   • Mixing Problems
   • Logistic Population Model
   • Falling Bodies with Air Resistance

First-Order Ordinary Differential Equations flashcards for Differential Equations

21 of 59 cards from the First-Order Ordinary Differential Equations deck — real questions with worked answers.

1. What is the order of a differential equation?

   The order is the order of the highest derivative appearing in the equation. For example, $\frac{d^{2}y}{dx^{2}} + \left(\frac{dy}{dx}\right)^{3} = 0$ is second order.

2. Define the degree of a differential equation.

   The degree is the power of the highest-order derivative, after the equation has been made polynomial (rational and radical free) in its derivatives. It is only defined when such a polynomial form exists.

3. Distinguish an ordinary differential equation (ODE) from a partial differential equation (PDE).

   An ODE contains derivatives with respect to a single independent variable, e.g. $\frac{dy}{dx}=y$. A PDE contains partial derivatives with respect to two or more independent variables, e.g. $\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$.

4. When is an ODE called linear?

   When it can be written as $a_{n}(x)y^{(n)}+a_{n-1}(x)y^{(n-1)}+\cdots+a_{1}(x)y'+a_{0}(x)y=g(x)$, i.e. $y$ and all its derivatives appear to the first power, with coefficients depending only on $x$ and no products of $y$ with its derivatives.

5. What makes a first-order ODE nonlinear? Give an example.

   It is nonlinear if the dependent variable or its derivative appears in a nonlinear way (power $\neq 1$, product, or nonlinear function). Example: $\frac{dy}{dx}=y^{2}$ or $\frac{dy}{dx}=\sin y$.

6. What is an explicit solution of an ODE versus an implicit solution?

   An explicit solution expresses the dependent variable directly, $y=\phi(x)$. An implicit solution is a relation $G(x,y)=0$ that defines one or more solutions without solving for $y$.

7. What is meant by a general solution and a particular solution of a first-order ODE?

   The general solution is a family containing an arbitrary constant $c$ representing all solutions; a particular solution is obtained by assigning a specific value to $c$ (e.g. via an initial condition).

8. What is a singular solution of a differential equation?

   A singular solution satisfies the ODE but cannot be obtained from the general solution by any choice of the arbitrary constant. It often arises as an envelope of the family of solution curves.

9. How do you verify that a function $y=\phi(x)$ is a solution of an ODE on an interval $I$?

   Substitute $\phi(x)$ and its derivatives into the equation; it is a solution if the equation reduces to an identity for every $x$ in $I$, and $\phi$ is defined and differentiable on $I$.

10. State a first-order initial-value problem (IVP).

    $\frac{dy}{dx}=f(x,y),\quad y(x_{0})=y_{0}$. We seek a solution on an interval containing $x_{0}$ that also satisfies the initial condition.

11. How does a boundary-value problem (BVP) differ from an initial-value problem?

    An IVP specifies all conditions at a single point $x_{0}$. A BVP specifies conditions at two (or more) different points, e.g. $y(a)=y_{0},\ y(b)=y_{1}$ for a second-order equation.

12. State the existence-and-uniqueness theorem for the IVP $y'=f(x,y),\ y(x_{0})=y_{0}$.

    If $f$ and $\frac{\partial f}{\partial y}$ are continuous on a rectangle $R$ containing $(x_{0},y_{0})$, then there exists a unique solution on some interval about $x_{0}$.

13. Give an example showing that continuity of $f$ alone does not guarantee uniqueness.

    $\frac{dy}{dx}=y^{1/3},\ y(0)=0$ has both $y=0$ and $y=\left(\tfrac{2}{3}x\right)^{3/2}$ as solutions; $\frac{\partial f}{\partial y}=\tfrac{1}{3}y^{-2/3}$ is discontinuous at $y=0$.

14. What is a mathematical model, and what are the basic steps in building one for a dynamic system?

    A mathematical model is an equation describing a system. Steps: identify variables and assumptions, express the rate of change (often via a physical law) as a differential equation, solve it, and validate against observations.

15. Write the general form of a first-order separable equation and its solution method.

    Form: $\frac{dy}{dx}=g(x)h(y)$. Separate: $\frac{dy}{h(y)}=g(x)\,dx$, then integrate both sides: $\int\frac{dy}{h(y)}=\int g(x)\,dx + c$.

16. Solve $\frac{dy}{dx}=\frac{x}{y}$ by separation of variables.

    $y\,dy = x\,dx \Rightarrow \frac{y^{2}}{2}=\frac{x^{2}}{2}+c_{1}$, giving $y^{2}-x^{2}=c$.

17. When solving separable equations, why must you check values where $h(y)=0$?

    Dividing by $h(y)$ can lose constant solutions $y=y_{0}$ where $h(y_{0})=0$; these equilibrium solutions must be added separately as possible singular solutions.

18. State the test for exactness of $M(x,y)\,dx+N(x,y)\,dy=0$.

    The equation is exact if and only if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ on a simply connected region.

19. For an exact equation $M\,dx+N\,dy=0$, how do you find the solution $F(x,y)=c$?

    Find $F$ with $F_{x}=M$ and $F_{y}=N$: integrate $F=\int M\,dx+g(y)$, then differentiate with respect to $y$ and match to $N$ to determine $g(y)$. Solution: $F(x,y)=c$.

20. Solve the exact equation $(2xy)\,dx+(x^{2}-1)\,dy=0$.

    Here $M_{y}=2x=N_{x}$, exact. $F=\int 2xy\,dx = x^{2}y+g(y)$; $F_{y}=x^{2}+g'(y)=x^{2}-1\Rightarrow g(y)=-y$. Solution: $x^{2}y - y = c$.

21. What is an integrating factor, and why is it used?

    An integrating factor $\mu(x,y)$ is a multiplier that converts a non-exact (or non-standard) equation into an exact or directly integrable one, so it can be solved.

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