Question Statement

Solve the differential equation:

$2y(x+1)dy=xdx$

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Background and Explanation

This problem involves solving a first-order ordinary differential equation using the method of separation of variables, where we rearrange the equation so that all terms involving $y$ are on one side and all terms involving $x$ are on the other side before integrating.

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Solution

We begin with the given differential equation:

$2y(x+1)dy=xdx$

Step 1: Separate the variables

To separate the variables, we divide both sides by $(x+1)$ to isolate terms involving $y$ on the left and terms involving $x$ on the right:

$2ydy=\frac{x}{x+1}dx$

Step 2: Integrate both sides

Now we integrate both sides of the equation:

$2\int ydy=\int \frac{x}{x+1}dx$

Step 3: Evaluate the left-hand side

Using the power rule for integration:

$2\cdot \frac{y^{1+1}}{1+1}=2\cdot \frac{y^2}{2}=y^2$

Step 4: Evaluate the right-hand side

For the integral $\int \frac{x}{x+1}dx$, we use algebraic manipulation to simplify the integrand. We rewrite the numerator by adding and subtracting 1:

$\frac{x}{x+1}=\frac{x+1-1}{x+1}=\frac{x+1}{x+1}-\frac{1}{x+1}=1-\frac{1}{x+1}$

Therefore:

$\int \frac{x}{x+1}dx=\int \left(1-\frac{1}{x+1}\right)dx=\int 1dx-\int \frac{1}{x+1}dx$

Evaluating these standard integrals:

$=x-\ln (x+1)+c$

where $c$ is the constant of integration.

Step 5: Combine results

Equating both sides, we obtain the general solution:

$y^2=x-\ln (x+1)+c$

This can also be expressed as:

$y=\pm \sqrt{x-\ln (x+1)+c}$

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Key Formulas or Methods Used

• Separation of variables: Rearranging differential equations so that each variable appears on only one side of the equation
• Power rule for integration: $\int y^{n}dy=\frac{y^{n+1}}{n+1}+C$ (for $n\ne -1$)
• Algebraic manipulation: $\frac{x}{x+1}=1-\frac{1}{x+1}$ (adding and subtracting technique)
• Logarithmic integration: $\int \frac{1}{x+a}dx=\ln |x+a|+C$

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Summary of Steps

1. Separate variables: Divide by $(x+1)$ to get $2ydy=\frac{x}{x+1}dx$
2. Integrate both sides: $2\int ydy=\int \frac{x}{x+1}dx$
3. Simplify left side: $2\cdot \frac{y^2}{2}=y^2$
4. Rewrite integrand: Express $\frac{x}{x+1}$ as $1-\frac{1}{x+1}$
5. Integrate right side: $\int \left(1-\frac{1}{x+1}\right)dx=x-\ln (x+1)+c$
6. State final answer: $y^2=x-\ln (x+1)+c$