Exam Papers

Differential Calculus and Differential Equations 2022 – BSc Computer Science Part 1

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Paper code: 13502
1502
B.Sc. (Computer Science) (Part 1)
Examination, 2022
Paper No. 1.2
Differential Calculus and Differential Equations

Time: Three Hours]
[Maximum Marks: 50


Note: Attempt five questions in all selecting one question from each Section. All questions carry equal marks.

Section-A

1. (a) If y=a\cos\left ( \log x \right       )+b\sin\left ( \log x \right ) show that x^{2}y_{2}+xy_{1}+y=0 and x^{2}y_{n+2}+\left ( 2n+1       \right )xy_{n+1}+\left ( n^{2}+1 \right )y_{n}=0.

    (b) Find (y_{n})_{0} when y =       \sin(a \sin^{-1}x.

2. (a) Expand log(1+x) with the help of Maclavrin’s theorem.

        (b) Expand sin x in powers of (x-?/2) by Taylor’s theorem.

    3. (a) Find the sub-tangent and subnormal at the point t on the cycloid.

      x=a(t+\sin t), y = a(1-\cos t)

          (b) Find the pedal equation of the parabola :

      y2 = 4a (x + a)

      4. (a) In the curve rm = am cos m? prove that :

      \frac{ds}{d\theta}=a       \sec^{\frac{m-1}{m}}n\theta

            (b) Find the following limit :

        \lim_{x \to 0}\frac{\sin x -x +       \frac{x^{3}}{6}}{x^{2}}

        Section-B

        5. (a) Find :

          \int \frac{dx}{x^{3}-1}

              (b) Find :

          \int \frac{x}{\left ( x-3 \right       )\sqrt{x+1}}dx

          6. (a) Show that :

            \int_{0}^{\pi/2}\frac{\sqrt{\sin       x}}{\sqrt{\sin x}+ \sqrt{\cos x}}dx = \frac{1}{4}\pi

                (b) From the definition of a definite integral as the limit of a sum evaluate:

            \int_{a}^{b}e^{x}dx

            7. (a) Solve the differential equation :

              \frac{dy}{dx} = e^{x-y} +       x^{2}e^{-y}

                  (b) Solve the differential equation :

              \frac{dy}{dx} + \frac{y}{x} =       x^{2}

              Section-C

              8. Solve the following differential equations :

                  (a)

                \frac{d^{2}y}{dx^{2}}+ \frac{dy}{dx} +       y = e^{x}

                    (b)

                \frac{d^{3}y}{dx^{3}}+ y = \cos       2x

                9. Solve the following differential equations :

                    (a)

                x^{2}\frac{d^{2}y}{dx^{2}}+2x\frac{dy}{dx}-20y       = (x+1)^{2}

                    (b)

                x^{2}\frac{d^{2}y}{dx^{2}}-3x\frac{dy}{dx}+4y       = 2x^{2}

                10. (a) Solve:

                [latex]\frac{dx}{dt}-y=t,       \frac{dy}{dt}+x=1[/latex]

                    (b) Solve:

                [latex]\frac{dx}{dt}+7x-y=0,       \frac{dy}{dt}+2x+jy=0[/latex]

                ……End……

                Lokesh Kumar

                Being EASTER SCIENCE's founder, Lokesh Kumar wants to share his knowledge and ideas. His motive is "We assist you to choose the best", He believes in different thinking.

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