Question
Solve the given differential equation. \[ 6 x d x+d y=0 \]
Ask by Barnett Bates. in the United States
Jan 19,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution to the differential equation \( 6x \, dx + dy = 0 \) is:
\[
y = -3x^2 + C
\]
where \( C \) is a constant.
Solution
To solve the differential equation:
\[
6x \, dx + dy = 0
\]
We can interpret this as:
\[
dy = -6x \, dx
\]
Now, integrate both sides:
\[
\int dy = \int -6x \, dx
\]
This yields:
\[
y = -3x^2 + C
\]
where \( C \) is the constant of integration.
**Solution:**
All solutions are given by y equals –3 x² plus a constant. Thus,
y = −3 x² + C
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Simplify this solution Beyond the Answer
To solve the differential equation \( 6x \, dx + dy = 0 \), we can rearrange it to isolate \( dy \): \[ dy = -6x \, dx. \] Next, we integrate both sides. The left side integrates to \( y \), and the right side integrates as follows: \[ \int dy = y + C, \] \[ \int -6x \, dx = -3x^2 + C_1. \] Combining the constants, we have: \[ y = -3x^2 + C. \] Thus, the general solution to the differential equation is \[ y = -3x^2 + C, \] where \( C \) is the constant of integration.
