Question

Find the general solution of differential equation \( ydx + (x - y^2)dy = 0 \).

Hint:

When a differential equation is in the form \( M(x, y)dx + N(x, y)dy = 0 \), first check if it's exact by testing if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \). If it is, solving is straightforward. If not, try rearranging it into a linear form like \( \frac{dy}{dx} + P(x)y = Q(x) \) or \( \frac{dx}{dy} + P(y)x = Q(y) \). Recognizing the form is the key to solving first-order differential equations quickly.

Answer 1

Step 1: Understanding the Concept:
This is a first-order differential equation. We can check if it's an exact differential equation or rearrange it into a linear differential equation form.
Step 2: Key Formula or Approach:
The given equation can be rewritten in the form of a linear differential equation: \( \frac{dx}{dy} + P(y)x = Q(y) \).
The steps to solve this are:
1. Rearrange the equation into the standard linear form.
2. Identify \( P(y) \) and \( Q(y) \).
3. Calculate the Integrating Factor (I.F.) using the formula: \( I.F. = e^{\int P(y) dy} \).
4. The general solution is given by: \( x \cdot (I.F.) = \int Q(y) \cdot (I.F.) dy + C \).
Step 3: Detailed Explanation or Calculation:
The given differential equation is:
\[ ydx + (x - y^2)dy = 0 \] Step 3.1: Rearrange the equation.
Divide the entire equation by \( dy \):
\[ y \frac{dx}{dy} + x - y^2 = 0 \] Move the \( -y^2 \) term to the right side:
\[ y \frac{dx}{dy} + x = y^2 \] Divide by \( y \) to get the standard linear form \( \frac{dx}{dy} + P(y)x = Q(y) \):
\[ \frac{dx}{dy} + \frac{1}{y}x = y \] Step 3.2: Identify \( P(y) \) and \( Q(y) \).
Comparing with the standard form, we have:
\( P(y) = \frac{1}{y} \) and \( Q(y) = y \).
Step 3.3: Calculate the Integrating Factor (I.F.).
\[ I.F. = e^{\int P(y) dy} = e^{\int \frac{1}{y} dy} \] \[ I.F. = e^{\ln|y|} = |y| \] Assuming \( y>0 \), we can take \( I.F. = y \).
Step 3.4: Find the general solution.
The solution is given by \( x \cdot (I.F.) = \int Q(y) \cdot (I.F.) dy + C \).
Substitute the values of \( I.F. \) and \( Q(y) \):
\[ x \cdot y = \int y \cdot y \, dy + C \] \[ xy = \int y^2 \, dy + C \] \[ xy = \frac{y^3}{3} + C \] This is the general solution. We can also express \( x \) in terms of \( y \):
\[ x = \frac{y^2}{3} + \frac{C}{y} \] Step 4: Final Answer:
The general solution of the given differential equation is \( xy = \frac{y^3}{3} + C \).