This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Questions & Step-by-step Solutions
1 item
Q
Q: Solve the differential equation dy/dx = 2y.
Solution: This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Steps: 8
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.
