Differential Equations
Matching Differential Equations with Solutions
Grade 12
Question:
<p><strong>Match Column-I (Differential equation) with Column-II (Solution/Integral curves):</strong></p><p>(A)
\[\frac{dy}{dx} = y^2 + \left(\frac{dy}{dx}\right)^3 \frac{d^3y}{dx^3} - 3\left(\frac{d^2y}{dx^2}\right)^2 = 0\]
</p><p>(B)
\[(2x - 10y)\frac{dy}{dx} + y = 0\]
</p><p>(C)
\[\left(\frac{dy}{dx}\right)^2 \frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} = 0\]
</p><p>(D)
\[(x^2y^2 - 1)dy + 2xy^3dx = 0\]
</p><p><strong>Solutions:</strong></p><p>(P) y = A_1 x^2 + A_2 x + A_3</p><p>(Q) x^2y^2 + 1 = cy</p><p>(R) (x+1)(1-y) = cy</p><p>(S) x = A_1 y^2 + A_2 y + A_3</p><p>(T) xy^2 = 2y^5 + c</p>
Step-by-Step Solution
Key Concept: Match each differential equation with its corresponding solution by verifying that the solution satisfies the given differential equation.
<p><strong>Matching:</strong></p><p>A → R: The differential equation
\[\left(\frac{dy}{dx}\right)^2 \frac{d^3y}{dx^3} - 3\left(\frac{d^2y}{dx^2}\right)^2 = 0\]
is satisfied by
\[(x+1)(1-y) = cy\]
</p><p>B → T: The differential equation
\[(2x - 10y)\frac{dy}{dx} + y = 0\]
is satisfied by
\[xy^2 = 2y^5 + c\]
</p><p>C → S: The differential equation
\[\left(\frac{dy}{dx}\right)^2 \frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} = 0\]
is satisfied by
\[x = A_1 y^2 + A_2 y + A_3\]
</p><p>D → Q: The differential equation
\[(x^2y^2 - 1)dy + 2xy^3dx = 0\]
is satisfied by
\[x^2y^2 + 1 = cy\]
</p>
Correct Answer: A→R; B→T; C→S; D→Q