Question

Consider the differential equation

• A. \(y=4x\)
• B. \(y=-4x^2+1\)
• C. \(y=1-2x^2\)
• D. \(y=\log(x)+4x\)

Hint:

For first-order differential equations of the form \[ \frac{dy}{dx}=f(x), \] direct integration gives the general solution. Use the initial condition to find the constant of integration.

Answer 1

The Correct Option is C

Step 1: Write the given differential equation.
\[ \frac{dy}{dx}+4x=0 \]

Step 2: Rearrange the equation.
Move \(4x\) to the right-hand side.
\[ \frac{dy}{dx}=-4x \]

Step 3: Integrate both sides.
Integrating with respect to \(x\), we get
\[ \int \frac{dy}{dx}\,dx = \int -4x\,dx \]

Step 4: Perform the integration.
\[ y = -4\left(\frac{x^2}{2}\right)+C \] \[ y=-2x^2+C \]

Step 5: Use the initial condition.
Given that
\[ y=1 \quad \text{at} \quad x=0 \] Substitute into the solution.
\[ 1=-2(0)^2+C \] \[ C=1 \]

Step 6: Write the particular solution.
Substituting \(C=1\), we obtain
\[ y=1-2x^2 \]

Step 7: Final conclusion.
Therefore, the required solution is
\[ \boxed{y=1-2x^2} \]
Hence, the correct answer is option (C).