02 | First Order First Degree Ordinary Differential Equation | Differential Calculus

Summary

The lecture introduces methods for solving first-order, first-degree ordinary differential equations. The equation’s form determines which solution method to use.

Definition

• First order means the highest derivative in the equation is a first derivative.
• First degree means that the highest-order derivative appears to the first power.

A first-order, first-degree equation may be written in forms such as [ \frac{dy}{dx}=f(x,y) ] or [ M\,dx+N\,dy=0. ]

Separation of variables

Rearrange the equation so that terms involving (y) appear with (dy), and terms involving (x) appear with (dx): [ f(y)\,dy=g(x)\,dx. ]

Then integrate both sides with respect to their respective variables. Include an arbitrary constant of integration; constants can be renamed or combined to simplify the final solution.

Linear substitution

If (x) and (y) occur together in a linear expression, such as (x+y), introduce a new variable for that expression. For example, let (z=x+y).

Differentiate the substitution, replace the original expression and its derivative in the equation, and check whether the resulting equation can be separated and integrated.

Homogeneous equations

Homogeneity can be identified by checking whether the terms have the same total degree, or whether the equation can be expressed in terms of a ratio such as (y/x).

Use the substitution (y=vx), which gives [ \frac{dy}{dx}=v+x\frac{dv}{dx}. ]

Substitute into the equation, simplify, and separate the (v)-terms from the (x)-terms. Integrate, then substitute (v=y/x) back into the result. If simplification instead gives a function of (x/y), the corresponding ratio substitution may be used.

Exact equations and other forms

Equations written in differential form, such as [ M\,dx+N\,dy=0, ] are mentioned, but exact differential equations and their solution method are reserved for the next lecture. The lecture also briefly alludes to other reducible forms without developing them in detail.

The presenter closes by directing viewers to the course’s Telegram channel for notes and other materials.

Speakers or sources featured

• One instructor/presenter delivers the lecture.
• No other speakers or external quoted sources are featured.