APSC 2113 Lectures 1-6

The first six lectures build the core framework for solving and interpreting ODEs: first classify an equation by order and linearity, understand what makes a valid solution and how initial conditions determine a particular one, then choose a method based on its structure. The main methods developed are separation of variables, direction fields and basic modeling (especially mixing problems), orthogonal trajectories, and homogeneous first-order equations using \(y=ux\). The later lectures transition to second-order linear equations, introducing Cauchy–Euler equations and constant-coefficient equations, where the roots of a characteristic/auxiliary equation determine the form of the solution—including distinct, repeated, and complex roots—along with linear independence, the Wronskian, and the decomposition \(y=y_c+y_p\) for nonhomogeneous equations.

What is an ordinary differential equation (ODE)?

An equation containing ordinary derivatives of one or more dependent variables with respect to a single independent variable.

How is the order of a differential equation determined?

The order is the highest derivative that appears in the equation.

What makes a differential equation linear?

The dependent variable and its derivatives appear only to the first power, are not multiplied together, and their coefficients depend only on the independent variable.

What is the standard form of a linear nth-order ODE?

a_n(x)y^(n) + … + a_1(x)y' + a_0(x)y = g(x).

What is the difference between a homogeneous and nonhomogeneous linear ODE?

Homogeneous means g(x)=0; nonhomogeneous means g(x)≠0.

What is a solution of an ODE?

A function that has the required derivatives and satisfies the differential equation on an interval.

What is a general solution?

A family of solutions containing arbitrary constants.

How many arbitrary constants does the general solution of an nth-order ODE generally contain?

n arbitrary constants.

What is a particular solution?

A specific solution obtained by assigning values to the arbitrary constants, often using initial conditions.

What is an initial-value problem (IVP)?

A differential equation together with conditions specified at one initial point.

What is a boundary-value problem (BVP)?

A differential equation together with conditions specified at boundary points.

What is the difference between an explicit and implicit solution?

An explicit solution has y isolated as a function of x; an implicit solution gives a relationship G(x,y)=0.

How do you verify that a function is a solution of an ODE?

Differentiate as needed, substitute into the ODE, simplify, and verify the equation holds on the stated interval.

What does the existence-and-uniqueness theorem for a first-order IVP state?

If f(x,y) and ∂f/∂y are continuous in a rectangle containing (x₀,y₀), then a unique solution exists on some interval around x₀.

What is separation of variables?

A method that rewrites a first-order ODE so all y terms are with dy and all x terms are with dx, then integrates.

What form is suited for separation of variables?

dy/dx = g(x)h(y).

What is the general separation-of-variables procedure?

Separate variables, integrate both sides, apply the initial condition, and check the resulting solution.

Why must you check for lost solutions when separating variables?

Dividing by a factor such as y can remove solutions for which that factor equals zero.

What is a direction field?

A graph showing small line segments whose slopes equal f(x,y) for an ODE y'=f(x,y).

What is the main modeling equation for a mixing problem?

Rate of change = rate in − rate out.

How is concentration calculated in a mixing problem?

Concentration = amount of substance / volume of solution.

How is the outgoing substance rate calculated in a mixing problem?

Outgoing liquid rate × current concentration.

What is the standard procedure for a mixing problem?

Define the amount, write rate in − rate out, express concentration using the current amount, solve the resulting ODE, and apply the initial condition.

What is an orthogonal trajectory?

A curve that intersects every curve in a given family at right angles.

How are slopes related for orthogonal trajectories?

Their slopes are negative reciprocals, when both slopes are finite and nonzero.

How do you find orthogonal trajectories?

Find the differential equation of the original family, replace the slope with its negative reciprocal, then solve.

What is a homogeneous function of degree n?

A function satisfying f(tx,ty)=t^n f(x,y).

What substitution is used to solve a first-order homogeneous ODE?

y=ux, with dy=u dx+x du.

Why does the substitution y=ux work for a homogeneous first-order ODE?

It converts the equation into a separable equation in u and x.

What is the main recognition pattern for a Cauchy-Euler equation?

Terms have matching powers of x, such as ax²y''+bxy'+cy=0.

What substitution is used for a Cauchy-Euler equation?

y=x^m.

What is the characteristic equation for ax²y''+bxy'+cy=0?

a m²+(b−a)m+c=0.

What is the solution form for two distinct real Cauchy-Euler roots m₁ and m₂?

y=C₁x^m₁+C₂x^m₂.

What is the solution form for a repeated Cauchy-Euler root m?

y=C₁x^m+C₂x^m ln x.

What is the solution form for complex Cauchy-Euler roots α±βi?

y=x^α[C₁cos(βln x)+C₂sin(βln x)].

What trial solution is used for a constant-coefficient homogeneous linear ODE?

y=e^(rx).

How do you solve the auxiliary equation for a constant-coefficient ODE?

Replace derivatives by powers of r, set the resulting polynomial equal to zero, and solve for its roots.

What is the solution form for distinct real roots r₁ and r₂?

y=C₁e^(r₁x)+C₂e^(r₂x).

What is the solution form for a repeated root r?

y=(C₁+C₂x)e^(rx).

What is the solution form for complex roots a±bi?

y=e^(ax)[C₁cos(bx)+C₂sin(bx)].

What is the general solution of a nonhomogeneous linear ODE?

y=y_c+y_p, where y_c is the complementary solution and y_p is any particular solution.

What is the complementary solution?

The general solution of the associated homogeneous equation.

What is linear independence for two solutions?

Neither solution is a constant multiple of the other; equivalently, C₁y₁+C₂y₂=0 only when C₁=C₂=0.

What is the Wronskian of two functions y₁ and y₂?

W=y₁y₂'−y₂y₁'.

What is a key method-selection checklist for these lectures?

Separable → separation of variables; first-order homogeneous-function form → y=ux; Cauchy-Euler → y=x^m; constant-coefficient linear homogeneous → y=e^(rx); known solution → reduction of order.