Differential Equations
Differential Equations
star_batch_jee_advanced_2025
Grade 12
Question:
Find the constant of integration by the general solution of the differential equation $(2x^2y - 2y^4)dx + (x^3 + 3x^3y)dy = 0$ if curve passes through $(1, 1)$.
Step-by-Step Solution
Key Concept: Non-exact equations can sometimes be made exact by dividing by an appropriate integrating factor related to the structure of the equation.
The differential equation $2x^2 ydx - 2y^3 dx + 2x^3 dy + 3xy^2 dy = 0$ is divided by $x^3y$ to obtain $2\frac{dx}{x} - \frac{2y^3}{x^3}dx + 2\frac{dx}{y} + \frac{3y^2}{x^2}dy = 0$. Rearranging into exact form and identifying groupings allows integration to find the solution in implicit form.
Correct Answer: 1