Question

Solve the equation explicitly. $$ y^{\prime}=\frac{y^{2}+2 x y}{x^{2}} $$

Short answer

In conclusion, the solution to the given differential equation $$y^{\prime}=\frac{y^{2}+2xy}{x^{2}}$$ is $$y(x) = \frac{e^{-\frac{1}{x} + C}}{1 + 2x} \operatorname{sgn}(1 + 2x)$$.

Step-by-step solution

Rearrange the Equation in a Separable Form

First, we need to rearrange the equation by multiplying both sides by \(x^{2}\) and then dividing by \((y^{2} + 2xy)\) to make it separable: $$ \frac{y'(x)}{y^{2}+2xy} = \frac{1}{x^2} $$ Now, it's in the form of \(\frac{dy}{y^2 + 2xy} = \frac{dx}{x^2}\), which separates the variables y and x.

Integrate Both Sides

Now that the equation is in the separable form, we'll integrate both sides: $$ \int\frac{dy}{y^2 + 2xy} = \int\frac{dx}{x^2} $$ For the left-hand side, we perform a substitution; let \(z = y(1 + 2x)\). Then, \(dz=(1+2x)dy+2ydx\). We have: $$ \int\frac{dy}{y^2 + 2xy} = \int\frac{1+2x}{z}dz $$ So the equation becomes: $$ \int\frac{1+2x}{z}dz = \int\frac{dx}{x^2} $$

Perform the Integration

Now we can perform the integration: $$ \int\frac{1+2x}{z}dz = \ln|z| + C_1 $$ and $$ \int\frac{dx}{x^2} = -\frac{1}{x} + C_2 $$ Now, we express the first equation in terms of y using our substitution \(z=y(1 + 2x)\): $$ \ln|y(1+2x)| + C_1 = -\frac{1}{x} + C_2 $$

Solve for y(x)

In this final step, we'll rearrange the equation to solve explicitly for y(x). First, let \(C = C_2 - C_1\): $$ \ln|y(1+2x)| = -\frac{1}{x} + C $$ Now, take the exponential of both sides: $$ |y(1+2x)| = e^{-\frac{1}{x} + C} $$ Finally, isolate y(x): $$ y(x) = \frac{e^{-\frac{1}{x} + C}}{1 + 2x} \operatorname{sgn}(1 + 2x) $$ Thus, we have obtained the explicit solution for y(x) that satisfies the given differential equation.

Key concepts

Separable Equations

Separable equations are a special type of differential equations that can be broken down such that each variable can be separated on either side of the equation. This makes it possible to deal with one variable at a time, which simplifies solving the equation.
In our problem, we utilize separation by rearranging the equation by moving terms around. We arrive at:

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

This neat separation allows us to focus on integrating each part individually, targeting the variables separately. This step is vital as it paves the way for the integration process that follows. Getting this separation right, especially rearranging terms without altering their meanings, is key to solving these types of equations effectively.

Integration Techniques

Integration techniques come into play immediately after separating variables. In this scenario, after simplifying the differential equation, we proceed by integrating both sides separately.
For the right side, we have the simpler term:

• \( \int \frac{dx}{x^2} \)

This is a straightforward integration resulting in \( -\frac{1}{x} + C_2 \).
Meanwhile, integrating the left side requires a substitution strategy. By letting \( z = y(1 + 2x) \), we make the integral manageable:

• \( \int \frac{1+2x}{z}dz \)

After solving, this yields \( \ln|z| + C_1 \). Techniques such as substitution are powerful tools in a mathematician's arsenal for transforming cumbersome integrals into simpler forms.

Explicit Solutions

An explicit solution refers to directly expressing one variable entirely in terms of the other. After performing integrations for both sides, we need to solve for \( y \) explicitly.
From our integration results, we derive:

• \( \ln|y(1+2x)| = -\frac{1}{x} + C \)

To solve for \( y \), we exponentiate both sides to remove the logarithm:

• \( |y(1+2x)| = e^{(-\frac{1}{x} + C)} \)

Finally, dividing by \( (1+2x) \), we isolate \( y \):

• \( y(x) = \frac{e^{(-\frac{1}{x} + C)}}{1 + 2x} \operatorname{sgn}(1 + 2x) \)

This expression of \( y \) entirely in terms of \( x \) represents an explicit solution, making further analysis or use of the solution straightforward and analytical.

Variable Separation

Variable separation is an essential method for solving differential equations where you rearrange the equation so that one variable is on each side. This carries particular importance when approaching complex equations that might initially appear inseparable.
The main strategy involves manipulating the equation until each side includes only one of the variables along with its differential:
For our example, after appropriately rearranging and simplifying, we get:

• \( \int \frac{dy}{y^2 + 2xy} = \int \frac{dx}{x^2} \)

This transformation highlights how the equation is approachable using simple integrals of single-variable functions. Ensuring correct separation allows us to apply known techniques such as integrations more effectively. Mastering variable separation simplifies dealing with complex differential equations, unlocking the path to resolution.