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Federal (FBISE) · Chapter test · Free

2nd Year Mathematics Chapter 4 Test04: Differential Equations

25 marks · 40 minutes · 16 questions. Covers: 04: Differential Equations.

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2nd Year Mathematics Chapter 4 Test

2nd Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 25Time: 40 minutes

Q1. Multiple choice questions (Attempt all · 5 × 1 = 5)

  1. The solution of the differential equation dy = dx is:
    1. y = x + c
    2. y = x² + c
    3. y² = x² + c
    4. y² = x + c
  2. y = 8 is a solution of the differential equation:
    1. dydx+8y=32\frac{dy}{dx}+ 8y = 32
    2. dydx+6y=32\frac{dy}{dx}+ 6y = 32
    3. dydx+5y=32\frac{dy}{dx}+ 5y = 32
    4. dydx+4y=32\frac{dy}{dx}+ 4y = 32
  3. The solution of cosx siny dx + sinx cosy = 0 is:
    1. sinx cosy = c
    2. cosx cosy = c
    3. cosx siny = c
    4. sinx siny = c
  4. The general solution of y²dy - x²dx = 0 is:
    1. x² - y² = c
    2. x³ + y³ = c
    3. x³ - y³ = c
    4. x² + y² = c
  5. The differential equation dydx=ex+y\frac{dy}{dx}= e^{x +y} has solution.
    1. e−x−y=ce^{-x -y}= c
    2. e−x+ey=ce^{-x} + e^y = c
    3. ex+ey=ce^x + e^y = c
    4. e−x+e−y=c−ye^{-x} + e⁻y = c^{-y}

Q2. Short questions (Attempt any 6 of 8 · 6 × 2 = 12)

  1. Solve the initial value problems. (x²+ 2y²)dx = xy dy ; y (1) = 1
  2. Solve the differential equation by separating the variable.xdydx=4yx\frac{dy}{dx} =4y
  3. Identify linear and non-linear different equation.d2ydx2+12xy=0\frac{d^2y}{dx^2} + 12xy = 0
  4. Identify linear and non-linear different equation.dydx=a\frac{dy}{dx} = a
  5. Check whether the function. f(x,y)=x2y+4f(x , y) = \frac{x}{2y}+ 4
  6. Eliminate the arbitrary constants from the following equation and form a differential equation of the lowest order: y = A sin (2x - B)
  7. Solve the homogeneous differential equations.5. dydx=y−xx+y\frac{dy}{dx} = \frac{y - x}{x + y}
  8. Solve the homogeneous differential equations.dydx=y2+yxx2\frac{dy}{dx} = \frac{y^2 + yx}{x^2}

Q3. Long questions (Attempt any 2 of 3 · 2 × 4 = 8)

  1. Verify that the indicated function is a solution of the given differential equation: 2y′ + y = 0 ;y=e−x2y= e^-\frac{x}{2}
  2. Verify y=x416 y = \frac{x^4}{16} that is a solution of the differential equation dydx−xy12=0\frac{dy}{dx} - xy^{\frac{1}{2}} = 0
  3. Verify that the indicated function is a solution of the given differential equation: x3dy−2 dx=0 x^3 dy - 2 \, dx = 0 ; y=−1x2+6 y = -\frac{1}{x^2} + 6