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Solving Homogeneous Differential Equations: Techniques and Examples
Solving Homogeneous Differential Equations: Techniques and Examples
Homogeneous differential equations are a common type of ordinary differential equations where the function and its derivatives can be written as a homogeneous function. This article focuses on the techniques and methods to solve such equations, particularly in the context of exact and separable differential equations. We will explore a specific example with a detailed step-by-step solution.
Introduction to Homogeneous Differential Equations
A homogeneous differential equation is an equation of the form:
[ frac{dy}{dx} Fleft(frac{y}{x}right) ]
where ( F ) is a function of the ratio ( frac{y}{x} ). To solve such equations, we can use a substitution method involving ( v frac{y}{x} ).
Solving the Given Example
Consider the differential equation:
[ frac{dy}{dx} frac{x^2y}{x^3 y^3} ]
This is a homogeneous differential equation. To solve it, we make the substitution ( y vx ) where ( v ) is a function of ( x ).
Step 1: Substitution and Simplification
Substitute ( y vx ) and ( frac{dy}{dx} v xfrac{dv}{dx} ): Substitute into the original equation:[ v xfrac{dv}{dx} frac{x^2(vx)}{x^3 (vx)^3} ]
Simplify the right side:
Right side:
[ frac{x^2(vx)}{x^3 v^3 x^3} frac{vx^3}{x^3(1 v^3)} frac{v}{1 v^3} ]
So, the equation becomes:
[ v xfrac{dv}{dx} frac{v}{1 v^3} ]
Step 2: Isolate the Derivative Term
Isolate ( xfrac{dv}{dx} ):
[ xfrac{dv}{dx} frac{v}{1 v^3} - v ]
Simplify the right side:
[ xfrac{dv}{dx} frac{v - v(1 v^3)}{1 v^3} frac{v - v - v^4}{1 v^3} frac{-v^4}{1 v^3} ]
So, the equation becomes:
[ xfrac{dv}{dx} frac{-v^4}{1 v^3} ]
Step 3: Separation of Variables
Separate variables to solve:
[ frac{1 v^3}{v^4} dv -frac{dx}{x} ]
Step 4: Integration
Integrate both sides:
[ int frac{1 v^3}{v^4} dv -int frac{1}{x} dx ]
On the left side, split the integrand:
[ int left( frac{1}{v^4} frac{v^3}{v^4} right) dv -ln|x| C ]
? 1 1 v 4 v 3 v 4 d v V V ? - ln | x | C
Evaluating the integral on the left:
∫ (1 - v 3 ) v 4 dv V V - ln x C
The integral evaluates to:
[ -frac{1}{3v^3} - ln|v| -ln|x| C ]
Distribute the negative signs and simplify:
[ frac{1}{3v^{-3}} - ln v ln x C ]
Convert the term ( frac{1}{3v^{-3}} ) to a more readable form:
[ v^{-3} frac{1}{v^3} ]
[ frac{1}{3}v^3 - ln v ln x C ]
Since ( v frac{y}{x} ), substitute back:
[ frac{1}{3}left(frac{y}{x}right)^3 - lnleft(frac{y}{x}right) ln x C ]
Multiplying through by ( 3x^3 ) for clarity:
[ y^3 - 3x^2lnleft(frac{y}{x}right) 3Cx^3 ]
Thus, the general solution is:
[ y^3 - 3x^2lnleft(frac{y}{x}right) 3Cx^3 ]
Conclusion
This article has detailed the process to solve a specific homogeneous differential equation. By understanding and applying the techniques of substitution and separation of variables, we can solve a wide range of such equations. The final solution provides useful insight into the behavior of solutions to differential equations.
Key Points
Homogeneous differential equations are equations that can be expressed in the form ( frac{dy}{dx} Fleft(frac{y}{x}right) ). The substitution ( y vx ) simplifies the equation, where ( v frac{y}{x} ). Separation of variables allows for the integration of both sides, leading to the solution.Keywords
homogeneous differential equation separable differential equation exact equation-
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