1(a)
Using Taylor's series, find y(0.4) where dy/dx = 1 + xy with y(0) = 2.

3 M

1(b)
Find the complementary function of

3 M

1(c)
Evaluate the following:

3 M

1(d)
Evaluate the following:

3 M

1(e)
Show that the following holds true:

4 M

1(f)
Using Euler's method, solve: dy/dx = x + y, y(0) =1. Find the value of y at x=1, taking h = 0.2

4 M

2(a)
Evaluate the following:

6 M

2(b)
Use Runge-Kutta method of fourth order to solve dy/dx=1/(x+y); y(0)=1. Find y(0.2) with h=0.1

6 M

2(c)
Solve the following:

8 M

3(a)
Solve the following:

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3(b)
Solve the following using variation of parameters.

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3(c)
Evaluate the following:

and hence deduce that:

and hence deduce that:

8 M

4(a)
Solve the following: (xy

^{3}+y)dx+2(x^{2}y^{2}+x+y^{4})dy=0
6 M

4(b)
Solve the following:

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4(c)
Solve the following:

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5(a)
In a circuit containing Inductance L, Resistance R, and Voltage E, the current i is L di/dt+Ri=E. Find i at time t. At t=0,i=0, L,R,E are constants.

6 M

5(b)
Change the order of integration:

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5(c)
Evaluate the following

over the volume of a sphere x

over the volume of a sphere x

^{2}+ y^{2}+ z^{2}= a^{2}
8 M

6(a)
Find the length of the parabola x

^{2}= 4y which lies inside the circle x^{2}+ y^{2}= 6y.
6 M

6(b)
Change into polar and evaluate:

6 M

6(c)
Evaluate the following

over the area of the triangle formed by x = 0, y = 0, x + y = 1.

over the area of the triangle formed by x = 0, y = 0, x + y = 1.

8 M

7(a)
Change the order of integration and evaluate

6 M

7(b)
Find the area outside the circle r=a and inside the cardioid r=a(1+cosθ)

6 M

7(c)
Find the volume common to the cylinders: x

^{2}+ y^{2}= a^{2}, x^{2}+ z^{2}= a^{2}
8 M

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