Question Statement

Solve the differential equation:

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

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Background and Explanation

This is a homogeneous differential equation because the right-hand side can be expressed purely as a function of the ratio $\frac{y}{x}$. To solve such equations, we use the substitution $y=ux$, which transforms the equation into a separable form that can be solved by direct integration.

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Solution

Step 1: Identify the equation type and substitute

The given differential equation is:

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

This is a homogeneous differential equation. We make the substitution:

$y=ux$

where $u$ is a function of $x$. Differentiating both sides with respect to $x$ using the product rule:

$\frac{dy}{dx}=\frac{du}{dx}\cdot x+u\cdot 1=x\frac{du}{dx}+u$

Step 2: Transform the differential equation

Substituting $y=ux$ and $\frac{dy}{dx}=x\frac{du}{dx}+u$ into equation (1):

$x\frac{du}{dx}+u=\frac{(ux{)}^2+(ux)\cdot x}{x^2}$

Simplifying the right-hand side:

$x\frac{du}{dx}+u=\frac{u^2{x}^2+u{x}^2}{x^2}=\frac{x^2(u^2+u)}{x^2}=u^2+u$

Step 3: Separate the variables

Subtracting $u$ from both sides:

$x\frac{du}{dx}=u^2+u-u=u^2$

Rearranging to separate variables:

$xdu=u^{2}dx$

$\frac{du}{u^2}=\frac{dx}{x}$

Step 4: Integrate both sides

Integrating both sides:

$\int u^{-2}du=\int \frac{dx}{x}$

Applying the power rule on the left and the logarithmic rule on the right:

$\frac{u^{-2+1}}{-2+1}=\ln x+c$

$\frac{u^{-1}}{-1}=\ln x+c$

$-\frac{1}{u}=\ln x+c$

Step 5: Back-substitute to find the final solution

Recall that $y=ux$, which means $u=\frac{y}{x}$. Substituting this into equation (2):

$-\frac{1}{\frac{y}{x}}=\ln x+c$

Simplifying:

$\boxed{-\frac{x}{y}=\ln x+c}$

Alternatively, solving for $y$ explicitly:

$y=\frac{-x}{\ln x+c}$

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Key Formulas or Methods Used

• Homogeneous substitution: $y=ux$ to convert $f(y/x)$ into a function of $u$
• Product rule for differentiation: $\frac{d}{dx}(ux)=u+x\frac{du}{dx}$
• Separation of variables: Rearranging to the form $f(u)du=g(x)dx$
• Power rule for integration: $\int u^{n}du=\frac{u^{n+1}}{n+1}$ (for $n\ne -1$)
• Logarithmic integration: $\int \frac{1}{x}dx=\ln |x|$

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Summary of Steps

1. Verify homogeneity: Confirm the equation can be written as $\frac{dy}{dx}=f\left(\frac{y}{x}\right)$
2. Substitute: Let $y=ux$, giving $\frac{dy}{dx}=u+x\frac{du}{dx}$
3. Simplify: Substitute into the original equation and cancel terms to get $x\frac{du}{dx}=u^2$
4. Separate variables: Rearrange to $\frac{du}{u^2}=\frac{dx}{x}$
5. Integrate: Evaluate $\int u^{-2}du=\int \frac{dx}{x}$ to get $-\frac{1}{u}=\ln x+c$
6. Back-substitute: Replace $u$ with $\frac{y}{x}$ to obtain the final solution $-\frac{x}{y}=\ln x+c$