8x+x In(x))/x3 (11 + 8x + x ln(x))/x2 none of the above, specify: Problem #4 Sol

Question

8x+x In(x))/x3 (11 + 8x + x ln(x))/x2 none of the above, specify: Problem #4 Solve the initial value problem as follows: dy dy to FOUR figures and present it below (12 points): (your numerical result for the value of x must 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) + y) 2 [-] z=4+ (4 z=(4 x + y) z=4x+y none of the above, specify: The initial value problem for z is as follows (mark a correct variant) (2 points): 2) =2-

Explanation / Answer

Put z = 4x+y

differentiate both sides with respect to x

=> dz/dx = 4+dy/dx

dy/dx +4+(4x+y)2 = 0

=> dz/dx + z2 = 0

=> dz/dx = -z2

=> dz/-z2 = dx

Integrate both sides

=> 1/z = x+c (c is constant of integration)

=> 1/(4x+y) = x+ c

When x = 0 , y= 1

1/(4*0+1) = 0+c

=> c = 1

1/(4x+y) = x + 1

=> 4x+y = 1/(1+x)

dy/dx +4+(4x+y)2 = 0

=> dy/dx +4+(1/(1+x)2 = 0

-4.04 +4+(1/(1+x)2) = 0

1/(1+x)2 = 0.04 = 4/100 = 1/25

(1+x)2 = 25 => 1+x = 5 or 1+x = -5 => x = 4 , x = -6

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