Convert the integral equation f (x) = x^2 - integral_0^x t f (t) dt into an ODE by...


Convert the integral equation

{eq}\displaystyle f (x) = x^2 - \int_0^x t f (t) \ dt {/eq}

into an ODE by differentiating both sides with respect to x. Now, find the general solution of the ODE. Only one of these solutions will solve the integral equation. Determine which by substituting {eq}x = 0 {/eq} into the above integral equation.

First-Order Linear Differential Equations:

An equation which can be written in the form {eq}y'+a(x)y=b(x) {/eq} is called a first-order linear differential equation for the unknown function {eq}y {/eq}. If multiplying this equation by {eq}K(x) {/eq} will put the equation into a form which can be easily integrated, then we say that {eq}K(x) {/eq} is an integrating factor for the equation. Any first-order linear differential equation {eq}y'+a(x)y=b(x) {/eq} has integrating factors of the form {eq}K(x)=e^{\int a(x) \, dx} {/eq}.

Answer and Explanation: 1

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Differentiating the equation {eq}f(x)=x^2-\int_0^x t f(t) \, dt {/eq} gives

{eq}\begin{align*} f'(x)&=2x-\frac{d}{dt}\int_0^x t f(t) \,...

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First-Order Linear Differential Equations


Chapter 16 / Lesson 3

Learn to define what a linear differential equation and a first-order linear equation are. Learn how to solve the linear differential equation. See examples.

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