Differential EquationsCollege3 topic explainers

Differential Equations course

Analytic solution methods for the most common ordinary differential equations.

Topics in this course

1.Separable Equations
A differential equation relates a function to its derivatives. It is separable when it can be written $\frac{dy}{dx} = g(x)h(y)$ , so that all the $y$ terms can be moved to one side and all the $x$ terms to the other: $\frac{dy}{h(y)} = g(x)\,dx$ . Integrating both sides then gives the solution, with one constant of integration.
- Divide by $h(y)$ only where $h(y) \ne 0$ ; constant solutions $h(y) = 0$ may need to be added back.
- One constant $C$ is enough after integrating both sides.
- Solve for $y$ explicitly when possible, then apply the initial condition.
Quick check: For $y = 3e^{x^2}$ , find $y(1)$ to two decimals.
$8.15$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study separable equations
2.Linear First-Order Equations
A first-order linear differential equation has the form $\frac{dy}{dx} + P(x)y = Q(x)$ . Unlike separable equations, the variables cannot generally be split, but an integrating factor $\mu(x) = e^{\int P(x)\,dx}$ makes the left side the derivative of a product, $\frac{d}{dx}(\mu y)$ .
- Write the equation in standard form, with coefficient 1 on $\frac{dy}{dx}$ , before finding $\mu$ .
- $\mu = e^{\int P\,dx}$ ; no constant is needed in this integral.
- After multiplying, the left side is always $(\mu y)'$ .
Quick check: For $y = 2 - 2e^{-2x}$ , what value does $y$ approach as $x \to \infty$ ?
$2$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study linear first-order equations
3.Second-Order Linear (Constant Coefficients)
A second-order linear equation with constant coefficients, $ay'' + by' + cy = 0$ , describes springs, pendulums and electrical circuits. Trying $y = e^{rx}$ turns it into the characteristic equation $ar^2 + br + c = 0$ , an ordinary quadratic whose roots determine the shape of every solution.
- Characteristic equation: replace $y''$ with $r^2$ , $y'$ with $r$ , $y$ with 1.
- Real distinct roots: $y = C_1 e^{r_1 x} + C_2 e^{r_2 x}$ .
- Complex roots produce sines and cosines; negative real part means damping.
Quick check: What is the sum of the roots of the characteristic equation of $y'' + 4y' + 3y = 0$ ?
$-4$
Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study second-order linear (constant coefficients)

Topics in this course

1.Separable Equations

2.Linear First-Order Equations

3.Second-Order Linear (Constant Coefficients)

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