What is the particular solution of the equation dy/dx = 2y with y(0) = 5?

What is the particular solution of the equation dy/dx = 2y with y(0) = 5?

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.
Step 4: Integrate both sides. The left side becomes ln|y| and the right side becomes 2x + C (where C is a constant).
Step 5: Write the result of the integration: ln|y| = 2x + C.
Step 6: Exponentiate both sides to solve for y: y = e^(2x + C).
Step 7: Rewrite e^(2x + C) as y = e^(2x) * e^C. Let C' = e^C, so y = C'e^(2x).
Step 8: The general solution is y = Ce^(2x), where C is a constant.
Step 9: Use the initial condition y(0) = 5 to find the specific value of C.
Step 10: Substitute x = 0 into the general solution: y(0) = Ce^(2*0) = C.
Step 11: Set C equal to 5 since y(0) = 5, so C = 5.
Step 12: The particular solution is y = 5e^(2x).

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