Question: Find the general solution of the differential equation dy/dx = 2y.
Options:
y = Ce^(2x)
y = 2Ce^x
y = Ce^(x/2)
y = 2x + C
Correct Answer: y = Ce^(2x)
Solution:
This is a separable equation. Integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Find the general solution of the differential equation dy/dx = 2y.
Practice Questions
Q1
Find the general solution of the differential equation dy/dx = 2y.
y = Ce^(2x)
y = 2Ce^x
y = Ce^(x/2)
y = 2x + C
Questions & Step-by-Step Solutions
Find the general solution of the differential equation dy/dx = 2y.
Step 1: Identify the differential equation. We have dy/dx = 2y.
Step 2: Recognize that this is a separable equation, meaning we can separate y and x.
Step 3: Rewrite the equation to separate variables: dy/y = 2 dx.
Step 4: Integrate both sides. The left side becomes ln|y| and the right side becomes 2x + C (where C is the constant of integration).
Step 5: Write the result of the integration: ln|y| = 2x + C.
Step 6: To solve for y, exponentiate both sides to eliminate the natural logarithm: |y| = e^(2x + C).
Step 7: Rewrite e^(2x + C) as e^(2x) * e^C. Let e^C be a new constant, which we can call C (since it is still a constant).
Step 8: Now we have |y| = C * e^(2x). Since C can be positive or negative, we can drop the absolute value: y = C * e^(2x).
Step 9: This is the general solution of the differential equation.
Separable Differential Equations – This concept involves equations that can be separated into two parts, one involving y and the other involving x, allowing for integration.
Integration of Natural Logarithm – Understanding how to integrate and manipulate logarithmic functions is crucial for solving the equation.
Exponential Functions – Recognizing that the solution involves an exponential function derived from the integration process.
Soulshift Feedback×
On a scale of 0–10, how likely are you to recommend
The Soulshift Academy?
