Higher Order Differential Equations Calculator

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Example

Created on 2024-06-20Asked by James Williams (Solvelet student)

Solve the third-order differential equation

y''' - 3y'' + 3y' - y = 0

Solution

To solve the third-order differential equation

y''' - 3y'' + 3y' - y = 0

: First, find the characteristic equation associated with the differential equation:

r^3 - 3r^2 + 3r - 1 = 0

Factor the characteristic equation:

(r - 1)^3 = 0

This gives a repeated root:

r = 1

The general solution to the differential equation is given by:

y(x) = (C_1 + C_2 x + C_3 x^2) e^{rx}

For

r = 1

y(x) = (C_1 + C_2 x + C_3 x^2) e^x

Therefore, the general solution to the differential equation

y''' - 3y'' + 3y' - y = 0

is:

y(x) = (C_1 + C_2 x + C_3 x^2) e^x

Where

C_1

C_2

, and

C_3

are arbitrary constants. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Luna Nguyen on Solvelet

1. Find the general solution to the third-order linear homogeneous differential equation

y''' - 2y'' + y' - y = 0

,2. Solve the initial value problem for the fourth-order linear nonhomogeneous differential equation

y''' + 4y'' = 3\sin(x)

with initial conditions

y(0) = y'(0) = 0

y''(0) = 1

, and

y'''(0) = 2

DefinitionA higher-order differential equation is a differential equation that contains derivatives of an unknown function of order higher than one. Equations which describe how dynamic systems and phenomena develop over time, and are used in physics, engineering, and applied mathematics. Any higher-order differential equation has to slug in an initial or boundary condition for each derivative of various order. But, this second-order differential equation dx2d2y+4dxdy+4y=0

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