(A) Given the differential equation: $x^2 + y^2 \frac{dy}{dx} = 4$. Rearranging the terms to separate the variables, we get: $y^2 dy = (4 - x^2) dx$.

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(A) Given the differential equation: $x^2 + y^2 \frac{dy}{dx} = 4$. Rearranging the terms to separate the variables, we get: $y^2 dy = (4 - x^2) dx$. Integrating both sides: $\int y^2 dy = \int (4 - x^2) dx$. This yields: $\frac{y^3}{3} = 4x - \frac{x^3}{3} + C_1$. Multiplying the entire equation by $3$: $y^3 = 12x - x^3 + 3C_1$. Let $3C_1 = C$, then the solution is: $x^3 + y^3 = 12x + C$.

Class 12 Mathematics Differential Equations Variable separable type differential equations

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The solution of $x^2 + y^2 \frac{dy}{dx} = 4$ is

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A
$x^3 + y^3 = 12x + C$
B
$x^2 + y^2 = 4x + C$
C
$x^3 - y^3 = 12x + C$
D
$x^3 + y^3 = 4x + C$

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