Differential Equations First-Order Ordinary Differential Equations Syllabus

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.

If $\frac{M_{y}-N_{x}}{N}$ depends only on $x$, what integrating factor makes $M\,dx+N\,dy=0$ exact?

$\mu(x)=\exp\!\left(\int \frac{M_{y}-N_{x}}{N}\,dx\right)$.

If $\frac{N_{x}-M_{y}}{M}$ depends only on $y$, what integrating factor makes $M\,dx+N\,dy=0$ exact?

$\mu(y)=\exp\!\left(\int \frac{N_{x}-M_{y}}{M}\,dy\right)$.

Write the standard form of a linear first-order ODE.

$\frac{dy}{dx}+P(x)\,y=Q(x)$, where $P$ and $Q$ are functions of $x$ only.

State the integrating factor for the linear equation $y'+P(x)y=Q(x)$.

$\mu(x)=e^{\int P(x)\,dx}$.

Give the general solution formula for $y'+P(x)y=Q(x)$.

With $\mu=e^{\int P\,dx}$, the solution is $y=\frac{1}{\mu}\left(\int \mu\,Q\,dx + c\right)$, since $\frac{d}{dx}(\mu y)=\mu Q$.

Solve $\frac{dy}{dx}+2y=e^{x}$ using an integrating factor.

$\mu=e^{2x}$; $\frac{d}{dx}(e^{2x}y)=e^{3x}$; $e^{2x}y=\tfrac{1}{3}e^{3x}+c$; so $y=\tfrac{1}{3}e^{x}+ce^{-2x}$.

Write the Bernoulli equation and state the substitution that linearizes it.

$\frac{dy}{dx}+P(x)y=Q(x)y^{n}$, $n\neq 0,1$. Substitute $u=y^{1-n}$, giving the linear equation $\frac{du}{dx}+(1-n)P(x)u=(1-n)Q(x)$.

Why are $n=0$ and $n=1$ excluded from the Bernoulli treatment?

For $n=0$ the equation $y'+Py=Q$ is already linear, and for $n=1$ it becomes $y'+(P-Q)y=0$, which is linear/separable. The substitution $u=y^{1-n}$ is only needed for other $n$.

Solve $\frac{dy}{dx}+y=y^{2}$ (a Bernoulli equation with $n=2$).

Let $u=y^{-1}$, so $u'-u=-1$. Then $u=1+ce^{x}$, giving $y=\dfrac{1}{1+ce^{x}}$.

What is a homogeneous first-order equation (in the sense of degree), and how is it solved?

An equation $\frac{dy}{dx}=f(x,y)$ where $f(tx,ty)=f(x,y)$; equivalently $M,N$ are homogeneous of the same degree. Substitute $y=vx$ (so $\frac{dy}{dx}=v+x\frac{dv}{dx}$), which yields a separable equation in $v$ and $x$.

A function $f(x,y)$ is homogeneous of degree $n$ means what?

$f(tx,ty)=t^{n}f(x,y)$ for all $t>0$. For example, $x^{2}+xy$ is homogeneous of degree $2$.

How do you handle $\frac{dy}{dx}=\frac{a_{1}x+b_{1}y+c_{1}}{a_{2}x+b_{2}y+c_{2}}$ when the lines are not parallel?

Shift the origin to the intersection point via $x=X+h,\ y=Y+k$ (with $h,k$ solving the two lines $=0$), removing the constants and reducing it to a homogeneous equation in $X,Y$.

Write the Riccati equation.

$\frac{dy}{dx}=P(x)+Q(x)y+R(x)y^{2}$, a quadratic-in-$y$ first-order nonlinear equation.

Given one particular solution $y_{1}$ of a Riccati equation, what substitution reduces it, and to what type?

Substitute $y=y_{1}+\frac{1}{u}$ (or $y=y_{1}+u$), which converts the Riccati equation into a linear (respectively Bernoulli) first-order equation in the new variable.

What does a direction field (slope field) of $y'=f(x,y)$ represent?

At each point $(x,y)$ it draws a short segment with slope $f(x,y)$; solution curves are tangent to these segments everywhere, giving a qualitative picture of solutions without solving.

What is an isocline in a direction field?

An isocline is the curve $f(x,y)=k$ along which all direction-field segments have the same slope $k$; plotting isoclines helps sketch the field efficiently.

Define an autonomous first-order differential equation.

One in which the independent variable does not appear explicitly: $\frac{dy}{dx}=f(y)$. Its slope depends only on $y$.

What are critical points (equilibria) of the autonomous equation $y'=f(y)$?

The constant solutions $y=c$ where $f(c)=0$. They correspond to the zeros of $f$ and appear as horizontal solution lines.

On a phase line, how do you classify a critical point as asymptotically stable, unstable, or semi-stable?

Stable (attractor): arrows point toward it on both sides. Unstable (repeller): arrows point away on both sides. Semi-stable: arrows point toward it on one side and away on the other.

Using the derivative test, when is a critical point $c$ of $y'=f(y)$ asymptotically stable?

If $f(c)=0$ and $f'(c)<0$, then $c$ is asymptotically stable; if $f'(c)>0$ it is unstable. ($f'(c)=0$ is inconclusive.)

Write the Picard iteration formula for the IVP $y'=f(x,y),\ y(x_{0})=y_{0}$.

Equivalent integral equation $y(x)=y_{0}+\int_{x_{0}}^{x} f(t,y(t))\,dt$; iterate $y_{n+1}(x)=y_{0}+\int_{x_{0}}^{x} f\big(t,y_{n}(t)\big)\,dt$ starting from $y_{0}(x)=y_{0}$.

What does the method of successive approximations (Picard iteration) establish theoretically?

Under a Lipschitz condition on $f$ in $y$, the iterates $y_{n}$ converge uniformly to the unique solution of the IVP, providing a constructive proof of existence and uniqueness.

Compute the first two Picard iterates for $y'=y,\ y(0)=1$.

$y_{0}=1$; $y_{1}=1+\int_{0}^{x}1\,dt=1+x$; $y_{2}=1+\int_{0}^{x}(1+t)\,dt=1+x+\frac{x^{2}}{2}$, approaching $e^{x}$.

State the growth/decay model and its solution.

$\frac{dP}{dt}=kP,\ P(0)=P_{0}$ gives $P(t)=P_{0}e^{kt}$. $k>0$ is exponential growth, $k<0$ is decay.

For radioactive decay with half-life $T$, relate the decay constant $k$ to $T$.

From $P(t)=P_{0}e^{kt}$ with $k<0$, $\tfrac12 P_{0}=P_{0}e^{kT}$ gives $k=-\frac{\ln 2}{T}$, i.e. $T=\frac{\ln 2}{|k|}$.

State Newton's law of cooling as a differential equation.

$\frac{dT}{dt}=k\,(T-T_{m})$, where $T$ is the object's temperature, $T_{m}$ the constant ambient temperature, and $k<0$ a constant.

Give the solution of Newton's law of cooling $\frac{dT}{dt}=k(T-T_{m}),\ T(0)=T_{0}$.

$T(t)=T_{m}+(T_{0}-T_{m})e^{kt}$; as $t\to\infty$, $T\to T_{m}$.

Set up the differential equation for a mixing (compartment) problem for the amount of salt $A(t)$.

$\frac{dA}{dt}=R_{in}-R_{out}=(\text{inflow rate}\times\text{inflow conc.})-(\text{outflow rate}\times\frac{A}{V(t)})$, where $V(t)$ is the current volume in the tank.

In a mixing problem, how is the tank volume $V(t)$ determined when inflow and outflow rates differ?

$V(t)=V_{0}+(r_{in}-r_{out})\,t$, where $r_{in},r_{out}$ are the volumetric flow rates; $V$ is constant only when $r_{in}=r_{out}$.

Write the logistic population model differential equation.

$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right)$, where $r$ is the intrinsic growth rate and $K$ the carrying capacity.

Give the solution of the logistic equation with $P(0)=P_{0}$.

$P(t)=\dfrac{K P_{0}}{P_{0}+(K-P_{0})e^{-rt}}$, so $P(t)\to K$ as $t\to\infty$.

What are the equilibrium solutions of the logistic equation and their stability?

$P=0$ (unstable) and $P=K$ (asymptotically stable). Populations with $0<P_{0}<K$ increase toward $K$; those above $K$ decrease toward $K$.

At what population size does the logistic model predict the fastest growth (maximum $\frac{dP}{dt}$)?

At the inflection point $P=\frac{K}{2}$, where $\frac{d^{2}P}{dt^{2}}=0$ and the growth rate is greatest.

Compare separable, linear, and exact first-order equations by their identifying form.

Separable: $y'=g(x)h(y)$ (variables split). Linear: $y'+P(x)y=Q(x)$ (first power of $y$, use $\mu=e^{\int P}$). Exact: $M\,dx+N\,dy=0$ with $M_{y}=N_{x}$ (potential $F$ exists).

How can you recognize and solve $\frac{dy}{dx}=f(ax+by+c)$?

Substitute $u=ax+by+c$; then $\frac{du}{dx}=a+b\,f(u)$, which is separable in $u$ and $x$.

What is the interval of validity (interval of definition) of a solution to an IVP?

The largest open interval containing $x_{0}$ on which the particular solution is defined, continuous, and satisfies the ODE; it may be smaller than the domain of $f$.

State the superposition-free general solution structure of a linear first-order equation in terms of complementary and particular parts.

$y=y_{c}+y_{p}$, where $y_{c}=ce^{-\int P\,dx}$ solves the homogeneous equation $y'+Py=0$ and $y_{p}$ is any particular solution of $y'+Py=Q$.

For the autonomous logistic-type harvesting model $\frac{dP}{dt}=rP(1-\frac{P}{K})-h$, what determines whether the population can survive?

Whether $f(P)=rP(1-\frac{P}{K})-h$ has real positive critical points; if the constant harvest $h$ exceeds the maximum growth $\frac{rK}{4}$, no equilibrium exists and the population declines to extinction.