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Find the general solution of the equation y' = 3x^2y.
Find the general solution of the equation y' = 3x^2y.
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Practice Questions
1 question
Q1
Find the general solution of the equation y' = 3x^2y.
y = Ce^(x^3)
y = Ce^(3x^3)
y = C/x^3
y = Cx^3
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This is a separable equation. Integrating gives y = Ce^(x^3).
Questions & Step-by-step Solutions
1 item
Q
Q: Find the general solution of the equation y' = 3x^2y.
Solution:
This is a separable equation. Integrating gives y = Ce^(x^3).
Steps: 8
Show Steps
Step 1: Recognize that the equation y' = 3x^2y is a separable differential equation.
Step 2: Rewrite the equation in a separable form: dy/y = 3x^2 dx.
Step 3: Integrate both sides: ∫(1/y) dy = ∫3x^2 dx.
Step 4: The left side integrates to ln|y|, and the right side integrates to x^3 + C, where C is the constant of integration.
Step 5: Write the equation as ln|y| = x^3 + C.
Step 6: Exponentiate both sides to solve for y: |y| = e^(x^3 + C).
Step 7: Rewrite e^(x^3 + C) as e^(x^3) * e^C. Let C' = e^C, which is a new constant.
Step 8: Thus, y = C'e^(x^3). Since C' can be any real number, we can write it as y = Ce^(x^3), where C is any constant.
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