Solving First-Order Differential Equations: y2dx x2dy

The given differential equation is y^2 dx x^2 dy. To solve this, we utilize the method of separation of variables to separate the variables on each side of the equation.

Step 1: Separation of Variables

Starting from the equation:

y^2 dx x^2 dy

We can rearrange it to separate the variables x and y as follows:

frac{dx}{x^2} frac{dy}{y^2}

Step 2: Integration

Next, we integrate both sides of the equation:

int frac{dx}{x^2} int frac{dy}{y^2}

The integral of the left side is:

-frac{1}{x} C_1

The integral of the right side is:

-frac{1}{y} C_2

Setting the integrals equal to each other, we get:

-frac{1}{x} -frac{1}{y} C

where C is the constant of integration. Simplifying, we get:

frac{1}{x} frac{1}{y} - C

or, solving for y in terms of x and C:

frac{1}{y} frac{1}{x} C

and thus:

y frac{1}{frac{1}{x} C}

Step 3: Verification and General Solutions

The given differential equation is separable, and through our steps, we obtain the general solution:

y frac{1}{frac{1}{x} C}

Particular solutions to the differential equation can also be checked. For example:

y x

y -frac{1}{x}

These solutions are consistent with our general solution. Additionally, by further manipulation, we can derive alternative forms, such as:

y tan left( -frac{1}{1x} C right)

Highlighting the versatility of the solution.

Conclusion

The equation y^2 dx x^2 dy is solved through the method of separation of variables, leading to the general solution y frac{1}{frac{1}{x} C}. Exploring alternative forms of the solution provides deeper insight into the behavior of this differential equation.