How do you find the solution to the differential equation dydx=cos⁡(x)y2 where y(π2)=0?

You can write:y2dy=cos⁡(x)dxand integrate;∫y2dy=∫cos⁡(x)dxWhich gives you:y33=sin⁡(x)+cy3=3sin⁡(x)+cNow we find the value of c;(0)3=3sin⁡(π2)+c0=3+cAnd c=-3your particular solution is then:y3=3sin⁡(x)−3

Question: "How do you find the solution to the..."

High School
Calculus and Analysis
Calculus 2
Applications of integrals

Blaine Jimenez

Answered question

2022-04-09

How do you find the solution to the differential equation

\frac{d y}{d x} = \frac{\cos (x)}{y^{2}}

where

y (\frac{π}{2}) = 0

Answer & Explanation

phoenixtreeaung

Beginner2022-04-10Added 19 answers

You can write:

y^{2} d y = \cos (x) d x

and integrate;

\int y^{2} d y = \int \cos (x) d x

Which gives you:

\frac{y^{3}}{3} = \sin (x) + c

y^{3} = 3 \sin (x) + c

Now we find the value of c;

{(0)}^{3} = 3 \sin (\frac{π}{2}) + c

0=3+c
And c=-3
your particular solution is then:

y^{3} = 3 \sin (x) - 3

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