Solve The Given Differential Equation By Using An Appropriate Substitution. The De Is Homogeneous. Y
When faced with a differential equation, especially a homogeneous differential equation involving the variable y, selecting the right substitution method can significantly simplify the solving process. In this article, we will explore how to solve a given homogeneous differential equation by using an appropriate substitution, focusing on the substitution \( y = vx \) or \( x = vy \). These substitutions help transform the original differential equation into a separable or more manageable form, making it easier to find the general solution.
Understanding Homogeneous Differential Equations
What is a Homogeneous Differential Equation?
A differential equation is called homogeneous if it can be expressed in the form:\[ \frac{dy}{dx} = F\left(\frac{y}{x}\right) \]
or equivalently, when the right-hand side can be written as a function of the ratio \( y/x \). Homogeneous equations are characterized by their scale invariance, meaning that replacing \( y \) with \( ky \) and \( x \) with \( kx \) preserves the form of the equation.
Key features of homogeneous differential equations:
- The right-hand side depends only on the ratio \( y/x \).
- They often appear in physics and engineering problems where proportional relationships are involved.
- They can be transformed into separable equations using substitution techniques.
Why Use Substitution for Homogeneous Equations?
Using substitution simplifies the process of solving homogeneous differential equations because it reduces the original equation to a separable form. Typically, the substitution involves expressing \( y \) as a multiple of \( x \) (or vice versa), which simplifies the dependence on the variables.
Common substitutions include:
- \( y = vx \), where \( v \) is a function of \( x \).
- \( x = vy \), when more appropriate.
In this article, we focus on the substitution \( y = vx \), which is most common for homogeneous equations involving \( y/x \).
Step-by-Step Solution Using the Substitution \( y = vx \)
1. Recognize the Homogeneous Equation
Identify if the differential equation is homogeneous by checking whether the right side can be expressed as a function of \( y/x \).For example:
\[ \frac{dy}{dx} = \frac{2xy + y^2}{x^2} \]
Rewrite the right side:
\[ \frac{2xy + y^2}{x^2} = \frac{2xy}{x^2} + \frac{y^2}{x^2} = 2 \frac{y}{x} + \left( \frac{y}{x} \right)^2 \]
Since the right side depends only on \( y/x \), the equation is homogeneous.
2. Make the Substitution \( y = vx \)
Set:\[ y = vx \]
where \( v \) is a function of \( x \). Then, differentiate \( y \):
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
This substitution transforms the original differential equation into an equation in terms of \( v \) and \( x \).
3. Rewrite the Differential Equation
Replace \( y \) and \( dy/dx \) in the original equation:\[ v + x \frac{dv}{dx} = F(v) \]
This often results in a separable differential equation in \( v \) and \( x \).
Using the example:
\[ v + x \frac{dv}{dx} = 2v + v^2 \]
Rearranged as:
\[ x \frac{dv}{dx} = 2v + v^2 - v = v + v^2 \]
which simplifies to:
\[ x \frac{dv}{dx} = v (1 + v) \]
4. Separate Variables and Integrate
Express the equation in a separable form:\[ \frac{dv}{v(1+v)} = \frac{dx}{x} \]
Now, perform partial fraction decomposition on the left:
\[ \frac{1}{v(1+v)} = \frac{A}{v} + \frac{B}{1+v} \]
Solve for \( A \) and \( B \):
\[ 1 = A(1 + v) + B v \]
Set \( v = 0 \):
\[ 1 = A(1) \Rightarrow A = 1 \]
Set \( v = -1 \):
\[ 1 = B (-1) \Rightarrow B = -1 \]
Thus:
\[ \frac{1}{v(1+v)} = \frac{1}{v} - \frac{1}{1+v} \]
Integrate both sides:
\[ \int \left( \frac{1}{v} - \frac{1}{1+v} \right) dv = \int \frac{1}{x} dx \]
which yields:
\[ \ln |v| - \ln |1+v| = \ln |x| + C \]
or, combining logs:
\[ \ln \left| \frac{v}{1+v} \right| = \ln |x| + C \]
Exponentiating both sides:
\[ \left| \frac{v}{1+v} \right| = K x \]
where \( K = e^{C} \).
Expressing the Solution in Terms of \( y \) and \( x \)
Recall that \( v = y / x \). Substitute back:
\[ \left| \frac{\frac{y}{x}}{1 + \frac{y}{x}} \right| = K x \]
Simplify numerator and denominator:
\[ \left| \frac{y / x}{(x + y) / x} \right| = K x \]
\[ \left| \frac{y}{x + y} \right| = K x \]
This implicit solution relates \( y \) and \( x \). To find an explicit form, proceed as follows:
\[ \frac{y}{x + y} = \pm K x \]
Cross-multiplied:
\[ y = \pm K x (x + y) \]
\[ y = \pm K x^2 + \pm K x y \]
Bring all \( y \) terms to one side:
\[ y - \pm K x y = \pm K x^2 \]
Factor \( y \):
\[ y (1 - \pm K x) = \pm K x^2 \]
Finally, solve for \( y \):
\[ y = \frac{\pm K x^2}{1 - \pm K x} \]
This expression provides the general solution to the original differential equation.
Additional Tips for Solving Homogeneous Differential Equations
1. Always Verify Homogeneity
Before applying substitution, confirm that the differential equation is homogeneous by checking if the right side can be written as a function of \( y/x \).2. Choose the Correct Substitution
- Use \( y = vx \) if the equation involves \( y/x \).
- Use \( x = vy \) if the equation involves \( x/y \).
3. Simplify and Integrate Carefully
Partial fractions and substitution are key to simplifying the integral. Always perform these steps carefully to avoid errors.4. Express the Final Solution Clearly
Depending on the context, you might want to solve explicitly for \( y \) or leave the solution in an implicit form.Conclusion
Solving homogeneous differential equations by using an appropriate substitution, such as \( y = vx \), is a powerful technique that simplifies complex equations into manageable forms. Recognizing the structure of the differential equation is crucial, as it guides the choice of substitution. Once the substitution is made, the problem becomes one of separation of variables and integration, leading to the general solution. Mastering this method not only enhances your problem-solving toolkit but also deepens your understanding of the fundamental properties of differential equations, especially those exhibiting scale invariance.
Whether you're working through academic problems or applying these techniques in engineering and physics, understanding how to solve homogeneous differential equations effectively is essential. With practice, identifying the right substitution and executing the steps systematically will become second nature, enabling you to tackle a wide range of differential equations with confidence.
