Final answer:
To solve the differential equation dx/dt = 2x where x(0) = 1, we can separate the variables by dividing both sides by x and dt, resulting in (1/x)dx = 2dt. We can integrate both sides to solve for x: ∫(1/x)dx = ∫2dt, which gives us ln|x| = 2t + C, where C is the constant of integration.
Step-by-step explanation:
To solve the differential equation dx/dt = 2x where x(0) = 1, we can separate the variables by dividing both sides by x and dt, resulting in (1/x)dx = 2dt. We can integrate both sides to solve for x: ∫(1/x)dx = ∫2dt, which gives us ln|x| = 2t + C, where C is the constant of integration. Next, we can exponentiate both sides to eliminate the natural logarithm: |x| = e^(2t + C). Since we started with a positive initial condition, x(0) = 1, we can drop the absolute value signs and solve for x: x = e^(2t + C).