Differential Equations Question - None

Question

Match Column-I (Differential equations) with Column-II (Solutions):

(A) Solution of differential equation: $$[3x^2y + 2xy - e^x(1 + x^2)]dx + (x^3 + x^2)dy = 0$$ is:

(B) Solution of differential equation: $$ydx - xdy - 3xy^2e^x dx = 0$$ is:

(C) Solution of differential equation: $$\frac{dy}{dx} = xy(x^2y^2 - 1)$$ is:

(D) Solution of differential equation: $$\frac{dy}{dx}(x^2y^3 + xy) = 1$$ is:

Solutions:

(P) y^2(x^2 + 1 + ce^{x^2}) = 1

(Q) (x^2 + x^3)y - xe^x = c

(R) \frac{x}{y} - \frac{3e^x}{2} = c

(S) \frac{1}{x} + 2e^{-y/2} = 2 - y + ce^{-y/2}

(T) \frac{2}{x} + e^{-y/2} = 1 - y + ce^{-y/2}

(where c is arbitrary constant)

Accepted Answer

Matching:

A → Q: The differential equation $$[3x^2y + 2xy - e^x(1 + x^2)]dx + (x^3 + x^2)dy = 0$$ is satisfied by $$(x^2 + x^3)y - xe^x = c$$

B → R: The differential equation $$ydx - xdy - 3xy^2e^x dx = 0$$ is satisfied by $$\frac{x}{y} - \frac{3e^x}{2} = c$$

C → P: The differential equation $$\frac{dy}{dx} = xy(x^2y^2 - 1)$$ is satisfied by $$y^2(x^2 + 1 + ce^{x^2}) = 1$$

D → S: The differential equation $$\frac{dy}{dx}(x^2y^3 + xy) = 1$$ is satisfied by $$\frac{1}{x} + 2e^{-y/2} = 2 - y + ce^{-y/2}$$

Question: