Question: Solve the first-order linear differential equation dy/dx = y/x.
Options:
y = Cx
y = Cx^2
y = C/x
y = C ln(x)
Correct Answer: y = Cx
Solution:
This is separable: dy/y = dx/x. Integrating gives ln|y| = ln|x| + C, thus y = Cx.
Solve the first-order linear differential equation dy/dx = y/x.
Practice Questions
Q1
Solve the first-order linear differential equation dy/dx = y/x.
y = Cx
y = Cx^2
y = C/x
y = C ln(x)
Questions & Step-by-Step Solutions
Solve the first-order linear differential equation dy/dx = y/x.
Step 1: Start with the equation dy/dx = y/x.
Step 2: Rewrite the equation to separate the variables: dy/y = dx/x.
Step 3: Integrate both sides: ∫(1/y) dy = ∫(1/x) dx.
Step 4: The left side becomes ln|y| and the right side becomes ln|x| + C (where C is a constant).
Step 5: Write the equation as ln|y| = ln|x| + C.
Step 6: Exponentiate both sides to eliminate the natural logarithm: |y| = e^(ln|x| + C).
Step 7: Simplify the right side: |y| = |x| * e^C. Let C' = e^C, so |y| = C'|x|.
Step 8: Remove the absolute value (considering C' can be positive or negative): y = Cx, where C is a constant.
First-order linear differential equations – These equations involve the first derivative of a function and can often be solved using separation of variables or integrating factors.
Separation of variables – A method used to solve differential equations by rearranging the equation to isolate the variables on different sides.
Integration of logarithmic functions – Understanding how to integrate functions involving logarithms is crucial for solving the equation after separation.
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