Differential Equations Basics

Differential Equations Basics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A differential equation is an equation that contains a derivative of an unknown function. They are fundamental tools in engineering and physics for describing how things change over time or space.

Key Definitions

• Differential Equation: An equation involving a function and its derivatives. Example: $\frac{dy}{dx} = 3x^2$
• Order: The highest derivative present. $\frac{d^2y}{dx^2} + 5\frac{dy}{dx} = x$ is a second-order DE.
• Degree: The highest power of the highest order derivative (when expressed as a polynomial in derivatives).
• Ordinary Differential Equation (ODE): Contains only one independent variable. $\frac{dy}{dx} = x + y$
• Partial Differential Equation (PDE): Contains partial derivatives with respect to multiple variables.

First-Order Differential Equations

Type 1: Variable Separable

If the equation can be written as $f(x)dx = g(y)dy$, we separate variables and integrate: $$\frac{dy}{dx} = xy$$ $$\frac{1}{y}dy = x \cdot dx$$ $$\int \frac{1}{y}dy = \int x \cdot dx$$ $$\ln|y| = \frac{x^2}{2} + C$$

Type 2: Linear First-Order DE

Form: $\frac{dy}{dx} + P(x)y = Q(x)$

The integrating factor is: $\mu(x) = e^{\int P(x)dx}$

Solution: $y \cdot \mu(x) = \int Q(x) \cdot \mu(x)dx + C$

Example: Solve $\frac{dy}{dx} + 2y = e^{3x}$

Here $P(x) = 2$, $Q(x) = e^{3x}$ $\mu(x) = e^{\int 2dx} = e^{2x}$

Multiply both sides by $e^{2x}$: $e^{2x}\frac{dy}{dx} + 2e^{2x}y = e^{5x}$ $\frac{d}{dx}(ye^{2x}) = e^{5x}$ $ye^{2x} = \int e^{5x}dx = \frac{e^{5x}}{5} + C$ $y = \frac{e^{3x}}{5} + Ce^{-2x}$

⚡ ECAT Tips

• Always include the constant of integration $C$ when integrating.
• Check whether your solution satisfies the original equation by differentiating it back.
• For word problems, translate the rate condition directly into a differential equation.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Types of First-Order DEs and Their Solutions

Exact Differential Equations

An equation $M(x,y)dx + N(x,y)dy = 0$ is exact if: $$\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$$

If exact, there exists a function $\psi(x,y)$ such that $d\psi = Mdx + Ndy = 0$: $\psi(x,y) = C$

Example: Check if $(2xy + y^2)dx + (x^2 + 2xy)dy = 0$ is exact.

$M = 2xy + y^2$, $\frac{\partial M}{\partial y} = 2x + 2y$ $N = x^2 + 2xy$, $\frac{\partial N}{\partial x} = 2x + 2y$

Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the equation is exact.

Integrate $M$ with respect to $x$: $\psi = \int (2xy + y^2)dx = x^2y + xy^2 + f(y)$

Differentiate with respect to $y$: $\frac{\partial \psi}{\partial y} = x^2 + 2xy + f’(y) = N = x^2 + 2xy$

So $f’(y) = 0$, thus $f(y) = C$

Therefore: $\psi(x,y) = x^2y + xy^2 = K$ Or: $xy(x + y) = K$

Homogeneous Differential Equations

A DE is homogeneous if $M(x,y)$ and $N(x,y)$ are both homogeneous functions of the same degree.

Solution method: Substitute $y = vx$, so $\frac{dy}{dx} = v + x\frac{dv}{dx}$

Example: Solve $\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$

$\frac{dy}{dx} = \frac{x^2 + y^2}{xy} = \frac{x}{y} + \frac{y}{x}$

Let $y = vx$, then $\frac{dy}{dx} = v + x\frac{dv}{dx}$: $v + x\frac{dv}{dx} = \frac{1}{v} + v$ $x\frac{dv}{dx} = \frac{1}{v}$ $v \cdot dv = \frac{1}{x}dx$ $\frac{v^2}{2} = \ln|x| + C$ Substituting back: $\frac{y^2}{2x^2} = \ln|x| + C$

Bernoulli’s Equation

Form: $\frac{dy}{dx} + P(x)y = Q(x)y^n$

Let $z = y^{1-n}$: $\frac{dz}{dx} + (1-n)P(x)z = (1-n)Q(x)$

Second-Order Linear DEs

General form: $a\frac{d^2y}{dx^2} + b\frac{dy}{dx} + cy = f(x)$

Complementary Function (CF): Solve $ay” + by’ + cy = 0$

Assume $y = e^{mx}$: $am^2 + bm + c = 0$

Three cases based on discriminant $b^2 - 4ac$:

1. Distinct real roots ($b^2 > 4ac$): $y_c = Ae^{m_1x} + Be^{m_2x}$
2. Equal real roots ($b^2 = 4ac$): $y_c = (A + Bx)e^{mx}$
3. Complex roots ($b^2 < 4ac$): $y_c = e^{\alpha x}(A\cos\beta x + B\sin\beta x)$

Particular Integral (PI): Based on $f(x)$ form:

• $f(x) = e^{kx}$ → try $y_p = Ce^{kx}$ (if $k$ not a root, multiply by $x$ if it is)
• $f(x) = x^n$ → try $y_p = Ax^n + Bx^{n-1} + …$
• $f(x) = \sin(kx)$ or $\cos(kx)$ → try $y_p = P\sin(kx) + Q\cos(kx)$

General Solution: $y = y_c + y_p$

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Advanced DE Solution Methods

Method of Undetermined Coefficients

For linear DEs with constant coefficients, guess a particular solution based on the form of $f(x)$.

$f(x)$ form	Trial particular solution
$e^{kx}$	$Ce^{kx}$ (multiply by $x$ if $k$ is root)
$x^n$	$A_nx^n + A_{n-1}x^{n-1} + … + A_0$
$\sin(kx)$ or $\cos(kx)$	$P\sin(kx) + Q\cos(kx)$
$e^{kx}\sin(mx)$	$e^{kx}(P\cos(mx) + Q\sin(mx))$

Example: Find the particular integral of $y” - 5y’ + 6y = e^{2x}$.

The auxiliary equation: $m^2 - 5m + 6 = 0$ $(m-2)(m-3) = 0$, so $m = 2, 3$

Since $k = 2$ is a root of the auxiliary equation, multiply by $x$: Try $y_p = Axe^{2x}$ $y_p’ = Ae^{2x} + 2Axe^{2x} = e^{2x}(A + 2Ax)$ $y_p” = 2Ae^{2x} + 2Ae^{2x} + 4Axe^{2x} = e^{2x}(4A + 4Ax)$

Substitute into LHS: $(4A + 4Ax) - 5(A + 2Ax) + 6(Ax) = e^{2x}$ $4A + 4Ax - 5A - 10Ax + 6Ax = e^{2x}$ $4A - 5A + (4A - 10A + 6A)x = e^{2x}$ $-A = e^{2x}$

This gives $A = -1$, so $y_p = -xe^{2x}$

Variation of Parameters

More general method for second-order linear DEs where method of undetermined coefficients fails.

For $y” + P(x)y’ + Q(x)y = R(x)$: Find CF: $y_c = u_1y_1 + u_2y_2$

Let $u_1’ = -\frac{y_2R}{W}$, $u_2’ = \frac{y_1R}{W}$

Where Wronskian $W = y_1y_2’ - y_1’y_2$

Particular solution: $y_p = -y_1\int\frac{y_2R}{W}dx + y_2\int\frac{y_1R}{W}dx$

Cauchy-Euler (Equidimensional) Equations

Form: $ax^2y” + bxy’ + cy = f(x)$

Try solution $y = x^m$: $m(m-1) + bm + c = 0$

Applications of Differential Equations

Population Growth (Malthusian): $$\frac{dP}{dt} = kP \Rightarrow P = P_0e^{kt}$$

Logistic Growth: $$\frac{dP}{dt} = kP(M - P) \Rightarrow P = \frac{M}{1 + Ae^{-kMt}}$$

Newton’s Law of Cooling: $$\frac{dT}{dt} = -k(T - T_s) \Rightarrow T = T_s + (T_0 - T_s)e^{-kt}$$

Electrical Circuits (RLC): For an RLC circuit: $L\frac{dI}{dt} + RI + \frac{q}{C} = E(t)$

Damped Oscillations: $m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = 0$

• Underdamped ($c^2 < 4mk$): $x = e^{-\alpha t}(A\cos\beta t + B\sin\beta t)$
• Critically damped ($c^2 = 4mk$): $x = (A + Bt)e^{-\alpha t}$
• Overdamped ($c^2 > 4mk$): $x = Ae^{m_1t} + Be^{m_2t}$

ECAT Engineering Context

In ECAT, differential equations often appear in:

• Calculating current in electrical circuits with resistors and capacitors
• Analysing spring-mass systems with damping
• Heat transfer problems (Newton’s law of cooling)
• Population dynamics and growth models

Working Tips

1. Always check initial/boundary conditions to find constants.
2. For non-homogeneous DEs, find the general solution of the homogeneous part first.
3. Verify your solution by substituting back into the original DE.
4. In engineering applications, the physical context suggests what type of solution to expect (oscillatory, decaying, growing).

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