Question: Solve 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. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Solve the differential equation dy/dx = 2y.
Practice Questions
Q1
Solve the differential equation dy/dx = 2y.
y = Ce^(2x)
y = 2Ce^x
y = Ce^(x/2)
y = 2x + C
Questions & Step-by-Step Solutions
Solve the differential equation dy/dx = 2y.
Step 1: Start with the differential equation dy/dx = 2y.
Step 2: Recognize that this is a separable equation, meaning we can separate y and x.
Step 3: Rewrite the equation as dy/y = 2 dx. This separates the variables.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy = ln|y|, and the right side becomes ∫2 dx = 2x + C, where C is the constant of integration.
Step 5: Now we have ln|y| = 2x + C.
Step 6: To solve for y, exponentiate both sides to remove the natural logarithm: y = e^(2x + C).
Step 7: Rewrite e^(2x + C) as y = e^(2x) * e^C. Let C' = e^C, which is also a constant.
Step 8: Finally, we can express the solution as y = C'e^(2x), where C' is a constant.
Separable Differential Equations – This concept involves equations that can be expressed as a product of a function of y and a function of x, allowing for separation of variables for integration.
Integration of Natural Logarithm – Understanding how to integrate functions involving natural logarithms and exponentials is crucial for solving the equation.
General Solution of Differential Equations – The solution includes a constant of integration (C), which represents the family of solutions to the differential equation.
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