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What is the solution of the differential equation dy/dx = y/x?
What is the solution of the differential equation dy/dx = y/x?
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Practice Questions
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Q1
What is the solution of the differential equation dy/dx = y/x?
y = Cx
y = Cx^2
y = C/x
y = C ln(x)
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This is separable. Integrating gives y = Cx.
Questions & Step-by-step Solutions
1 item
Q
Q: What is the solution of the differential equation dy/dx = y/x?
Solution:
This is separable. Integrating gives y = Cx.
Steps: 8
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Step 1: Start with the differential equation dy/dx = y/x.
Step 2: Recognize that this equation is separable, meaning we can separate the variables y and x.
Step 3: Rewrite the equation as dy/y = dx/x. This separates the variables.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy and the right side becomes ∫(1/x) dx.
Step 5: The integral of 1/y is ln|y| and the integral of 1/x is ln|x|. So, we have ln|y| = ln|x| + C, where C is the constant of integration.
Step 6: Exponentiate both sides to eliminate the natural logarithm. This gives us |y| = e^(ln|x| + C).
Step 7: Simplify the right side. e^(ln|x| + C) can be rewritten as |x| * e^C. Let C' = e^C, so |y| = C'|x|.
Step 8: Remove the absolute value by considering the constant C' can be positive or negative. Thus, y = Cx, where C is a constant.
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