Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
For the differential equation $(3x + 2y^2)dx + 2x(2x + 3y^2)dy = 0$
on simplification it reduces to $2(xy^3)d(xy^3) + d(x^3y^3) = 0$
solution is $x^2y^6 + x^3y^3 = c$
on simplification it reduces to $2(x^2y)d(x^2y) + d(x^2y^3) = 0$
solution is $x^2y^4 + x^2y^3 = 0$
Step-by-Step Solution
Key Concept: Recognize that $(3x + 2y^2)dx + 2x(2x + 3y^2)dy = 0$ can be regrouped as exact differentials by identifying terms that form $d(xy^3)$ and $d(x^3y^3)$, then factored as $2(xy^3)d(xy^3) + d(x^3y^3) = 0$ to obtain the solution $x^2y^6 + x^3y^3 = c$.
Given $3xydx + 4x^2dy + 2y^3dx + 6xy^2dy = 0$, regroup as $3x^2y\,dx + 4x^3\,dy + 2d(y^3) + d(x^3y^3) = 0$. These combine into exact differentials: $d(x^3y^3) + 2d(y^3) = 0$, yielding the solution $x^2y^6 + x^3y^4 = c$.
Correct Answer: 1,2