Solution of the differential equation \(\displaystyle\frac{dy}{dx}+\frac{3x}{2y}=0\) with y(1) = 1 is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2y^2+3x^2=5\)
The equation is variables separable:
\( \displaystyle \frac{dy}{dx}=-\frac{3x}{2y} \)
\(2y\,dy=-3x\,dx\)
Integrating both sides:
\( \displaystyle y^2=-\frac{3x^2}{2}+C \)
\(2y^2+3x^2=2C\)
Using y(1) = 1, that is x = 1, y = 1:
\(2(1)+3(1)=5=2C\)
So the solution is \(2y^2+3x^2=5\).
Tip: Option (3) also passes through (1, 1), but it has the coefficients of \(x^2\) and \(y^2\) swapped: it gives \(\displaystyle\frac{dy}{dx}=-\frac{2x}{3y}\), not \(\displaystyle-\frac{3x}{2y}\). Checking the point alone is not enough.