Question

Problem #4 Solve the initial value problem as follows: dy dy +4+ (4 x +y) Then determine the positive number r such that - -4.04. Round-off the value of this positive number x to FOUR figures and present it below (12 points): your mumerical result for the ae ust be written here) Also, you must provide some intermediate results obtained by you while solving the problem above: 1) The substitution used to solve the differential equation is as follows (mark a correct variant) (2 points): □z=4+ (4 x + y) 2 none of the above, specify: 2) The initial value problem for z is as follows (mark a correct variant) (2 points): dx dx dx dz ах none of the above, specify 3) The value of y as the function of ax is given by the formula that is equivalent to the formula as follows (mark a correct variant) (2 points): 2 (1+x+x2) 1+2x 1-4x-4x2 1+x V 2(1-x-2x2) y1+ none of the above, specify e value of as the function of x is given by the formula that is equivalent to the formula as follows (mark a correct variant) (2 points): dy=-2(3-4x-4x2) dx (1+2x)2 dy5(3+4x-4x2) dx (1+2が dy=-2 (1 +x+4x2) 5 + 8 x + 4x2 dy dx none of the above, specify:

Answer 1

(1)

Correct option:

(2)

Given Differential Equation is:

                 (1)

dy

Put

                                         (2)

Differentiating (2), we get:

$\frac{\mathrm{d} z}{\mathrm{d} x} = 4 + \frac{\mathrm{d} y}{\mathrm{d} x}$                                  (3)

Substituting (2) & (3), equation (1)becomes:

+ ~2-0

i.e.,

$\frac{\mathrm{d} z}{\mathrm{d} x} = - z^{2}$                      (4)

The Initial Condition:

$y\left.\begin{matrix} _ \end{matrix}\right|_{x=0}=1$                   (5)

Substituting (2), equation (5) becomes:

$z\left.\begin{matrix} _ \end{matrix}\right|_{x=0}=1$                           (6)

So,

Correct option:

$\frac{\mathrm{d} z}{\mathrm{d} x} = - z^{2}$,     $z\left.\begin{matrix} _ \end{matrix}\right|_{x=0}=1$

(3)

From (4), we get:

dz dr

Integrating both sides, we get:

da C

i,e,,

$\frac{1}{z}=x+C$                          (7)

Substituting (6), equation (7) becomes:

$C=1$

Substituting, (7) becomes"

$\frac{1}{z}=x+1$

Substituting (2), we get:

$\frac{1}{4x+y}=x+1$

i.e.,

                (8)

4 422 u=1-11-

So,

Correct option:

4 422 u=1-11-

(4)

Substituting (2), equation (8) becomes:

$\frac{\mathrm{d} y}{\mathrm{d} x}=-\frac{5+8x+4x^{2}}{(x+1)^{2}}$

So,

Correct option:

$\frac{\mathrm{d} y}{\mathrm{d} x}=-\frac{5+8x+4x^{2}}{(x+1)^{2}}$