Separable Differential Equations, Math Notes, TechAmbitionX - Compiled By Bilal Ahmad Khan AKA Mr. BILRED
Separable Differential Equations
Compiled By: Bilal Ahmad Khan AKA Mr. BILRED
What Are Separable Differential Equations?
A separable differential equation is a first-order differential equation that can be rewritten so that all terms involving y are on one side and all terms involving x are on the other.
Where Are They Used?
- Physics: Radioactive decay, Newton’s Law of Cooling
- Biology: Population growth, spread of diseases
- Engineering: Electrical circuits, fluid dynamics
- Finance: Interest rate models
Important Points About Separable Differential Equations
- Separable differential equations are always first-order equations (i.e., they only involve
dy/dxand no higher derivatives). - They cannot directly be used for second-order or higher-order differential equations unless a substitution reduces them to a separable form.
- These equations appear frequently in physics, biology, and engineering, modeling growth, decay, and fluid flow.
- After separation of variables, integration is required on both sides to find the solution.
- Initial conditions help determine the specific constant for a unique solution.
How to Solve a Separable Differential Equation?
Let’s solve an example step by step:
Example:
Given equation:
dy/dx = xy
Step 1: Separate Variables
Rearrange to move y terms to one side and x terms to the other:
dy/y = x dx
Step 2: Integrate Both Sides
∫(dy/y) = ∫ x dx
ln |y| = x²/2 + C
Step 3: Solve for y
Exponentiate both sides:
y = e^(x²/2 + C)
Since e^C is a constant, we rewrite:
y = C' e^(x²/2)
Step 4: Apply Initial Condition (If Given)
If y(0) = 3, substitute x = 0 and y = 3:
3 = C' e⁰ ⇒ C' = 3
Final Solution:
y = 3e^(x²/2)
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