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: Read the equation.
The equation is $\dfrac{dy}{dx} + 4x = 0$ with the condition $y=1$ at $x=0$.

Step 2: Isolate the derivative.
\[ \frac{dy}{dx} = -4x \]

Step 3: Integrate.
Integrating both sides, \[ y = -2x^2 + C \]

Step 4: Fix the constant.
Put $x=0,\ y=1$ to get $1 = 0 + C$, so $C = 1$.

Step 5: Answer.
The solution is \[ \boxed{y = 1 - 2x^2} \]