Find the general solution to the homogeneous second-order differential equation.

{eq}\displaystyle y''-4y'+13y=0 {/eq}

## Question:

Find the general solution to the homogeneous second-order differential equation.

{eq}\displaystyle y''-4y'+13y=0 {/eq}

General Solution of the Second Order Differential Equation:

An equation of the form {eq}{P_0}{x^n}\dfrac{{{d^n}y}}{{d{x^n}}} + {P_1}{x^{n - 1}}\dfrac{{{d^{n - 1}}y}}{{d{x^{n - 1}}}} + ... + {P_n}y = X {/eq} is called homogeneous linear differential equation. Where {eq}{P_0} , {P_1} , ...,{P_n} {/eq} are constants and {eq}X {/eq} is a function of {eq}x {/eq} only. The general solution of the second order differential equation is the solution which has two independent arbitrary constants.

## Answer and Explanation: 1

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**Given:**

- The homogeneous second order differential equation is {eq}y'' - 4y' + 13 = 0 {/eq}.

The symbolic form of the ODE is {eq}\left( {{D^2} - 4D...

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Chapter 10 / Lesson 16The quadratic formula is a useful aid in solving problems. Explore the quadratic formula in a simplified way and apply it to multiple solutions, as well as learn to identify when there is no solution.