Tribute to Euler

Calculus Level 4

If $\dfrac d{dx} \left( \dfrac {dy}{dx} \right) = \sqrt y$ , under the condition $y=1$ , and $\dfrac{dy}{dx} = \dfrac2{\sqrt3}$ at $x=0$ , find $y$ when $x=2\sqrt3$ .

The answer is 16.

1 solution

Aayush Patni
Apr 2, 2016

Let y=f(x)

f''(x) = √(f(x)

Multiplying both sides of above equation by f'(x) and then integrating One gets

(f'(x)^2)/2 = 2/3 *(f(x)^3/2)

By using initial conditions one finds that the constant of integration=0

We have f'(x) = (2/√3)f(x)^3/4

dy/dx = (2/√3)y^3/4

Thus y^-3/4 dy = (2/√3) dx

Integrating both sides one gets

4y^1/4 =( 2/√3)*x + c

It is given that y=1 at x=0.

By putting the relation in above equation we get c=4

4y^1/4 = (2/√3)x + 4

Put x=2√3 to get y=16

hahahha it's just like you knew the text i took the question from

Benjamin ononogbu - 5 years, 2 months ago

Nope. I definitely didn't. BTW from which book did u take this question???

Aayush Patni - 5 years, 2 months ago

it's a text written by H.K DASS

Benjamin ononogbu - 5 years, 2 months ago

Tribute to Euler

The answer is 16.

1 solution

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