Integral Equation
Solve this integral equation.
Age
16 to 18
Challenge level
Problem
Solve the following integral equation for $f(x)$: $$\int_0^x f(t)\,dt = 3f(x)+k\,,$$ where $k$ is a constant.
Getting Started
Differentiate both sides of the equation.
Student Solutions
Thank you for your solutions to Daniel and Ben (no schools given) and to Rajiv from the International School of Seychelles and Shaun from Nottingham High School.
The integral equation is: $$\int_0^x f(t)\,dt = 3f(x)+k,\quad
\quad(\star)$$ where $k$ is a constant. Differentiating both sides
of $(\star)$ gives $$f(x) = 3f'(x)$$ If there is a solution of
$(\star)$ it must be of the form $$f(x) = Ae^{x/3},$$ for some
constant $A$. We check to see whether or not this is a
solution.
For $f(x)=Ae^{x/3}$ we have $$\int_0^x Ae^{t/3}\,dt =
\Big[3Ae^{t/3}\Big]_0^x = 3Ae^{x/3}-3A.$$ Thus $f(x)=Ae^{x/3}$ is a
solution if and only if $A=-k/3$. The unique solution is $$f(x) =
{-k\over 3} e^{x/3}.$$
Teachers' Resources
A differential equation involves functions and derivatives.
Integral equations involve functions and integrals.
In this example you have an integral equation. First you have to differentiate both sides. Then you have to solve a differential equation where you should be able to spot the solution immediately. Then you have to check that you have found a solution by substituting the function you have found into the integral equation.
In this example you have an integral equation. First you have to differentiate both sides. Then you have to solve a differential equation where you should be able to spot the solution immediately. Then you have to check that you have found a solution by substituting the function you have found into the integral equation.