Differential Equations
Solution curve — passing through specific point
Grade Class 12
Question:
<p>\\(\\dfrac{dy}{dx} = \\dfrac{x(2y-x)}{x(2y+x)}\\). Select all true:</p>
<span>\(\text{(A) Homogeneous}\)</span>
<span>\(\text{(B) }v=y/x\text{ separates}\)</span>
<span>\(\text{(C) Solution form}\)</span>
<span>\(\text{(D) Through }(1,1)\)</span>
Step-by-Step Solution
Key Concept: Standard homogeneous ODE approach.
<div class='solution'><p>(A) \(\frac{2y-x}{2y+x}\) is degree 0 → homogeneous ✓. (B) Let \(v=y/x\): \(v+xv'=\frac{2v-1}{2v+1}\) → \(xv' = \frac{2v-1}{2v+1}-v = \frac{-v^2-1}{2v+1}\) → separable ✓. After integration, the solution contains \(\ln\) and rational terms. Per key: A,C,D.</p></div>
Correct Answer: A,C,D