Solve the separable differential equation: dy/dx = 2x * y for y > 0 with initial condition y(0) = 3.
Answer and explanation
Correct answer: C. y = 3 * e^(x^2)
- Step 1: Separate variables: (1/y) dy = 2x dx.
- Step 2: Integrate both sides: integral (1/y) dy = integral 2x dx => ln(y) = x^2 + C1.
- Step 3: Exponentiate: y = C e^(x^2). Apply initial condition y(0) = 3: 3 = C e^0 => C = 3. Thus, y = 3 * e^(x^2).
Answer
y = 3 * e^(x^2).
Why not the others: 'y = 3 * e^(2x)' integrates 2x as 2x instead of x^2; 'y = e^(x^2) + 2' applies the constant additively instead of multiplicatively; 'y = x^2 + 3' integrates dy/dx without separating the dependent variable y.
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