A differential equation is an equation involving at least one derivative of an unknown function. Sol

Question

A differential equation is an equation involving at least one derivative of an unknown function.

Solution terminology

A solution is a function, or family of functions, that satisfies the differential equation on a stated interval.

General and particular solutions

Term	Meaning
General solution	A family containing one or more arbitrary constants, such as $y=\mathrm{F}(x)+C$.
Initial condition	A specified value such as $y(x_0)=y_0$, used to determine an arbitrary constant.
Particular solution	The member of the family obtained after applying the given condition or conditions.

First-order direct integration

If the derivative is already a function of $x$ alone, integrate with respect to $x$.

Direct-integration template

\[\frac{\mathrm{d}y}{\mathrm{d}x}=\mathrm{p}(x)\quad\Longrightarrow\quad y=\int \mathrm{p}(x)\,\mathrm{d}x+C\]

The arbitrary constant $C$ is essential. Use an initial condition only after obtaining the general solution.

Enrichment — outside the H2 syllabus

\[\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=\mathrm{p}(x)\Longrightarrow\frac{\mathrm{d}y}{\mathrm{d}x}=\int \mathrm{p}(x)\,\mathrm{d}x+C\Longrightarrow y=\iint \mathrm{p}(x)\,\mathrm{d}x\,\mathrm{d}x+Cx+D\]

Accepted Answer

$\frac{\mathrm{d}y}{\mathrm{d}x}=f(x)\Rightarrow y=\int f(x)\,\mathrm{d}x+C$ $\frac{\mathrm{d}^2y}{\mathrm{d}x^2}=f(x)\Rightarrow y=\int\!\int f(x)\,\mathrm{d}x\,\mathrm{d}x+C_1x+C_2$ $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{f(x)}{g(y)}\Rightarrow\int g(y)\,\mathrm{d}y=\int f(x)\,\mathrm{d}x+C$ $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{g(y)}{f(x)}\Rightarrow\int\frac{1}{g(y)}\,\mathrm{d}y=\int\frac{1}{f(x)}\,\mathrm{d}x+C$ $\frac{\mathrm{d}y}{\mathrm{d}x}=g(y)f(x)\Rightarrow\int\frac{1}{g(y)}\,\mathrm{d}y=\int f(x)\,\mathrm{d}x+C$

Differential Equations and Direct Integration

Differential Equations and Direct Integration

Video Solution:

Solution:

Solution locked