Fundamentals & Definitions
What is meant by the order of a differential equation?
The highest order derivative present in the equation.
Classify: dydx=x2y\frac{dy}{dx} = x^2 ydxdy=x2y
Separable differential equation.
Name the method used for separable equations.
Separation of variables.
Which DE model describes cooling of objects?
Newton’s Law of Cooling.
Which DE models pollution diffusion?
First-order transport equations.
Define a first-order differential equation.
A differential equation involving only the first derivative of the dependent variable.
Classify: dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy+Py=Q
Linear first-order differential equation.
What method is used for non-exact equations?
Integrating factor method.
Which type of DE models population growth?
First-order separable differential equation.
Which sustainability system uses first-order DEs for energy optimization?
Smart grids.
Distinguish between general solution and particular solution.
General solution contains arbitrary constants; particular solution is obtained using initial conditions.
Identify the type:
(2xy+y2)dx+x2dy=0(2xy + y^2)dx + x^2 dy = 0(2xy+y2)dx+x2dy=0
Exact differential equation.
What substitution is used in homogeneous equations?
y=vxy = vxy=vx or v=yxv = \frac{y}{x}v=xy
RC circuits are modeled using which DE type?
First-order linear differential equations.
Epidemic spread models are based on what DE type?
Coupled first-order differential equations.
What is an initial value problem (IVP)?
A differential equation with specified values of the function at a given point.
What makes a differential equation homogeneous?
When it can be expressed as a function of yx\frac{y}{x}xy or both terms are of same degree.
State the integrating factor for a linear DE
dydx+Py=Q\frac{dy}{dx} + Py = Qdxdy+Py=Q
IF=e∫PdxIF = e^{\int P dx}IF=e∫Pdx
Battery discharge in EVs follows which DE structure?
First-order dynamic decay models.
Water resource sustainability relies on what DE principle?
Flow balance and transport rate equations.
Define degree of a differential equation.
The power of the highest order derivative when the equation is free from radicals and fractions.
A DE is non-exact. What classification step comes next?
Find an integrating factor.
Why is method selection critical in solving DEs?
Wrong classification leads to mathematically invalid solutions.
Why are first-order DEs ideal for real-time engineering systems?
They model rate-based dynamic changes in physical systems.
Why are DEs critical for achieving SDGs?
They model dynamic systems for prediction, optimization, and control.