Q1. The differential equation \((3a^2x^2 + by\cos x)\,dx + (2\sin x - 4ay^2)\,dy = 0\) is exact if:

Option (A): a = 3, b = 2 (Correct)
Option (B): a = 2, b = 3
Option (C): a = 3, b = 4
Option (D): a = 2, b = 5

Q2. If \(2 - Px + Qx^2 = 0\), then a particular integral is:

Option (A): y = x²
Option (B): y = 1/x (Correct)
Option (C): y = e^x
Option (D): y = e^{-x}

Q3. Let \(W(y_1, y_2, y_3)\) be the Wronskian. Then:

Option (A): \(W(y_1, y_2, y_3) = 0\) (Correct)
Option (B): Functions are linearly dependent
Option (C): Functions are linearly independent (Correct)
Option (D): None of these

Q4. The problem has:

Option (A): A unique solution
Option (B): No solution
Option (C): Finite number of solutions
Option (D): Infinitely many solutions (Correct)

Q5. Consider the differential equation \(x^2y'' - 2xy' + 2y = 6\) with \(y(0)=3\) and \(y'(0)=1\). Then:

Option (A): \(y(x) = c x^2 + x + 3\) is not a solution
Option (B): \(y(x) = c x^2 + x + 3\) is a unique solution
Option (C): \(y(x) = c x^2 + x + 3\) is a solution but not unique (Correct)
Option (D): None of these

Q6. Let \(y_p\) be the particular solution of \((D^3 + 3D^2 + 2D)y = x^2 + 4x + 8\). Then \(y_p\) is:

Option (A): A polynomial of degree 3 with integer coefficients
Option (B): A polynomial of degree 2 with integer coefficients
Option (C): A polynomial of degree 3 with rational coefficients (Correct)
Option (D): A polynomial of degree 4 with real coefficients

Q7. Let \(y\) be the solution of a differential equation. Then possible values of constants are:

Option (A): Option 1
Option (B): Option 2 (Correct)
Option (C): Option 3
Option (D): Option 4

Q8. Consider the differential equation with boundary conditions \(y(0)=0\) and \(y(1)=1\). The complete solution is:

Option (A): Parabola (Correct)
Option (B): Hyperbola
Option (C): Circle
Option (D): None of these

Q9. The critical point (0, 0) for the system \(x'(t)=x+3y\), \(y'(t)=3x+2y\) is a:

Option (A): Stable saddle point
Option (B): Unstable saddle point (Correct)
Option (C): Unstable node
Option (D): None of these

Q10. What is the solution of the equation?

Option (A): \(\log\left(\frac{y}{x}\right) - \frac{1}{x} = c\)
Option (B): \(\log x - \frac{x}{y} = c\) (Correct)
Option (C): \(\log\left(\frac{x}{y}\right) - \frac{1}{x} = c\)
Option (D): \(\log x + \frac{x}{y} = c\)

Q10. What is the solution of the equation?

Option (A): \(\log\left(\frac{y}{x}\right) - \frac{1}{x} = c\)
Option (B): \(\log x - \frac{x}{y} = c\) (Correct)
Option (C): \(\log\left(\frac{x}{y}\right) - \frac{1}{x} = c\)
Option (D): \(\log x + \frac{x}{y} = c\)