Homogeneous Differential Equation Calculator

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Example

Created on 2024-06-20Asked by Scarlett Perez (Solvelet student)

Solve the homogeneous differential equation

\frac{dy}{dx} = \frac{x^2 + y^2}{xy}

Solution

To solve the homogeneous differential equation

\frac{dy}{dx} = \frac{x^2 + y^2}{xy}

: Rewrite the equation in the form:

\frac{dy}{dx} = \frac{x^2 + y^2}{xy} = \frac{x^2}{xy} + \frac{y^2}{xy} = \frac{x}{y} + \frac{y}{x}

Let

v = \frac{y}{x}

, hence

y = vx

. Then:

\frac{dy}{dx} = v + x\frac{dv}{dx}

Substitute

y = vx

into the original equation:

v + x\frac{dv}{dx} = \frac{x}{vx} + v

v + x\frac{dv}{dx} = \frac{1}{v} + v

Subtract

v

from both sides:

x\frac{dv}{dx} = \frac{1}{v}

Separate variables:

v dv = \frac{1}{x} dx

Integrate both sides:

\int v \, dv = \int \frac{1}{x} \, dx

\frac{v^2}{2} = \ln|x| + C

Multiply through by 2:

v^2 = 2 \ln|x| + C'

Since

v = \frac{y}{x}

\left(\frac{y}{x}\right)^2 = 2 \ln|x| + C'

y^2 = x^2 (2 \ln|x| + C')

Therefore, the solution to the homogeneous differential equation is:

y^2 = x^2 (2 \ln|x| + C')

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Samuel Carter on Solvelet

1. Solve the homogeneous differential equation

(x^2 + y^2) \, dx - xy \, dy = 0

,2. Determine whether the functions

y_1(x) = e^{2x}

and

y_2(x) = xe^{2x}

are linearly independent solutions of the homogeneous differential equation

y'' - 4y' + 4y = 0

DefinitionDescription: A homogeneous differential equation is a differential equation with a homogenous function of order n defined such is the derivative of a function y of the continuous image of n. Symmetry and uniformity are the key to obtaining solutions to these equations, often through methods that actually take advantage of this feature. Homogeneous differential equations have applications in physics and engineering to represent systems with uniform properties. For example: y′′−2y′+y=0 is a homogeneous differential equation because all of the terms involve the same dependent variable y or its derivatives.

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