what is implicit solution in dy/dx=y^1/2/(x^2+1)(x^2+4)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to separate the variables multiplying by `(dx)/sqrty`   both sides such that:

`(dy)/(sqrt y) = (dx)/((x^2+1)(x^2+4))`

Notice that both factors`x^2+1`  and `x^2 + 4`  do not have real roots, hence you should use partial fraction decomposition such that:

`1/((x^2+1)(x^2+4)) = (ax+b)/(x^2+1) + (cx+d)/(x^2+4)`

Bringing the terms to a common denominator yields:

`1 = ax^3 + 4ax + bx^2 + 4b + cx^3 + cx + dx^2 + d`

`1 = x^3(a+c) + x^2(b+d) + x(4a+c) + 4b + d`

Equating coefficients of like powers yields:

`a+c = 0`

`b+d = 0`

4a+c = 0

`4b+d = 1`

Subtracting the second equation from the fourth yields:

`4b+d-b-d=1 =gt 3b=1 =gt b = 1/3 =gt d=-1/3`

Subtracting the first equation from the third yields:

`4a+c-a-c = 0 =gt 3a = 0 =gt a=c=0`

Hence, after decomposition, the fraction `1/((x^2+1)(x^2+4)) `  looks like:

`1/((x^2+1)(x^2+4)) = (1/3)(1/(x^2+1) - 1/(x^2+4))`

You need to find the solution to equation `(dy)/(sqrt y) = (dx)/ou((x^2+1)(x^2+4))`  hence you need to integrate both sides such that:

`int (dy)/(sqrt y) = int (dx)/((x^2+1)(x^2+4))`

`2sqrty = (1/3)int (dx)/(x^2+1) - (1/3)int (dx)/(x^2+4)`

`2sqrty = (1/3)arctan x- (1/6)arctan x + c`

`2sqrty = (1/6)arctan x + c =gt sqrty = (1/12)arctan x + c`

You need to raise to square both sides to find y such that:

`y = ((1/12)arctan x)^2 + c`

Hence, the general solution to differential equation is `y = ((1/12)arctan x)^2 + c.`

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