Quick Answer – What is a First-Order Linear Differential Equation?
A first-order linear ordinary differential equation is of the form dy/dx + P(x)y = Q(x). It is solved by multiplying both sides by an integrating factor IF = e∫P(x)dx, which makes the left side an exact derivative, leading to y = (1/IF) ∫ IF·Q(x) dx + C. This technique is fundamental in JEE Advanced and engineering mathematics.
Key Formula & Method
Standard form: dy/dx + P(x)y = Q(x)
Integrating Factor (IF) = e∫ P(x) dx
Solution: y·IF = ∫ IF·Q(x) dx + C
An initial condition y(x0) = y0 allows determination of the constant C.
Solved Examples
Example 1: dy/dx + (2/x)y = x2
IF = e∫ 2/x dx = e2 ln x = x2
Solution: y·x2 = ∫ x4 dx = x5/5 + C ⇒ y = x3/5 + C/x2
Example 2: dy/dx + y = ex, y(0)=2
IF = e∫1 dx = ex, y·ex = ∫ e2x dx = (1/2)e2x + C
Using y(0)=2: 2·1 = 1/2 + C ⇒ C=3/2 ⇒ y = (1/2)ex + (3/2)e-x
Frequently Asked Questions
Q1. How do I enter functions like sin(x) or e^x?
Type sin(x), cos(x), tan(x), e^(x) or exp(x), log(x) for natural log. Use * for multiplication and ^ for powers.
Q2. Does the calculator handle any P(x) and Q(x)?
It uses symbolic integration for common functions. For complex functions, it may fall back to numeric approximation of the integrals, but covers typical JEE/NCERT functions (polynomials, exponential, trig, log).
Q3. What if I don't provide an initial condition?
The solution will include an arbitrary constant C (shown as +C). You can later substitute your own condition.
Q4. Can it solve non-linear first-order ODEs?
No, this tool is strictly for linear equations of the form dy/dx + P(x)y = Q(x). Bernoulli or separable equations require other methods.
Q5. How precise is the numerical integration?
When analytical integration is not possible, a simple numerical quadrature is used with moderate accuracy. For exact results, provide integrable functions.