Question:
The integral equation
\[
y(x)=x+\int_0^3 \cos(x-t)y(t)\,dt
\]
is a:
The integral equation
\[
y(x)=x+\int_0^3 \cos(x-t)y(t)\,dt
\]
is a:
Show Hint
Fixed limits give Fredholm integral equation, while variable upper limit gives Volterra integral equation.
Updated On: May 19, 2026
- Volterra integral equation of first kind
- Singular integral equation
- Fredholm integral equation
- Volterra integral equation of second kind
Show Solution
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The Correct Option is C
Solution and Explanation
Concept:
Integral equations are classified according to their limits and the position of the unknown function.
Step 1: Observe the given integral equation. \[ y(x)=x+\int_0^3 \cos(x-t)y(t)\,dt \]
Step 2: Check the limits of integration.
The lower limit and upper limit are fixed constants: \[ 0 \quad \text{and} \quad 3 \] When both limits are constants, the integral equation is of Fredholm type.
Step 3: Check the unknown function.
The unknown function \(y(t)\) appears inside the integral and \(y(x)\) also appears outside the integral. Thus, it is a Fredholm integral equation of second kind. Since the option says Fredholm integral equation, that is the correct choice. \[ \therefore \text{Correct Answer is (C)} \]
Integral equations are classified according to their limits and the position of the unknown function.
Step 1: Observe the given integral equation. \[ y(x)=x+\int_0^3 \cos(x-t)y(t)\,dt \]
Step 2: Check the limits of integration.
The lower limit and upper limit are fixed constants: \[ 0 \quad \text{and} \quad 3 \] When both limits are constants, the integral equation is of Fredholm type.
Step 3: Check the unknown function.
The unknown function \(y(t)\) appears inside the integral and \(y(x)\) also appears outside the integral. Thus, it is a Fredholm integral equation of second kind. Since the option says Fredholm integral equation, that is the correct choice. \[ \therefore \text{Correct Answer is (C)} \]
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