Engineering Mathematics - III(DIFFERENTIAL CALCULUS)

The solution of the differential equation y″ − 3y′ + 2y = e^3x

1. C₁e^x + C₂e^2x + e^3x/2
2. C₁e^−x + C₂e^−2x + e^3x/2
3. C₁e^−x + C₂e^2x + e^3x/2
4. None of these

The solution of p + q = z is

1. f(xy, y log z) = 0
2. f(x + y, y + log z) = 0
3. f(x − y, y − log z) = 0
4. None of these

The series 1 − (1/(1!)²)(x/2)² + (1/(2!)²)(x/2)⁴ − (1/(3!)²)(x/2)⁶ + … is

1. J_1/2(x)
2. xJ_1/2(x)
3. xJ₀(x)
4. None of these

If ∫₋₁¹ P_n(x) dx = 2, then n is

1. 0
2. 1
3. 2
4. None of these

The particular solution of y″ + 9y′ = sin 3x is

1. −(x/6) cos 3x
2. (x/6) cos 3x
3. −(3x/2) cos 3x
4. (3x/2) cos 3x

The differential equation y″ + (s³ sin x)⁵y′ + y = cos x³ is

1. Homogeneous
2. Non-linear
3. Second order linear
4. Non-homogeneous with constant coefficient

The magnitude of the directional derivative of the function f(x, y) = x² + 3y² in a direction normal to the circle x² + y² = 2, at the point (1,1) is

1. 4√2
2. 5√2
3. 7√2
4. 9√2

If R = xi + yj + zk is the position vector, then div R is

1. 1
2. 3
3. −3
4. −2

The degree of the homogeneous function f(x, y) = (x^7/3 + y^7/3)/(x + y) is

1. 4/5
2. 4/3
3. 7/3
4. None of these

If f(x, y) = tan⁻¹ y/x + sin⁻¹ x/y, then x∂f/∂x + y∂f/∂y =

1. 2
2. 1
3. 0
4. −1

If y^1/m + y^−1/m = 2x, prove that
(i) (x² − 1)y₂ + xy₁ − m²y = 0
(ii) (x² − 1)y_n+2 + (2n + 1)xy_n+1 + (n² − m²)y_n = 0

If p² = a² cos²θ + b² sin²θ, prove that p + d²p/dθ² = a²b²/p³

Use Taylor’s theorem to expand f(x, y) = x^y in powers of (x − 1) and (y − 1).

Find the dimension of the rectangular box, open at the top, of maximum capacity whose surface area is 432 sq.cm.

If u = tan⁻¹(y²/x), prove that x² ∂²u/∂x² + 2xy ∂²u/∂x∂y + y² ∂²u/∂y² = −sin²u · sin 2u

If by the substitution u = x² − y², v = 2xy, f(x, y) = θ(u, v), show that ∂²f/∂x² + ∂²f/∂y² = 4(x² + y²)(∂²θ/∂u² + ∂²θ/∂v²)

Verify Stoke’s theorem for F = (2x − y)i − yz²j − y²zk over the upper half surface x² + y² + z² = 1, bounded by its projection on the xy −plane.

If the directional derivative of φ = ax²y + by²z + cxz² at the point (1,1,1) has maximum magnitude 15 in the direction parallel to the line (x − 1)/2 = (y − 3)/(−2) = z, find the values of a, b, c

Solve (x² + y²)(x dx + y dy) = a²(x dy − y dx)

Solve p(p + x) = y(x + y)

Solve (2y log y + y − x)dy = y dx

Solve, by the method of variation of parameters, y″ − 2y′ + y = e^x log x.

Solve x² d²y/dx² − 3x dy/dx + y = log x · (sin(log x) + 1)/x

Prove that J_m(x) = (1/π) ∫₀^π cos(mθ − x sin θ) dθ, where ’n’ is positive integer

Solve the equation : (z² − 2yz − y²)p + (xy + zx)q = xy − zx.

Solve the equation : (x − y)(px − qy) = (p − q)²