Solving A Differential Equation With An Integral

[@cameronspalding9792]

x/y = y’ implies y*y’ = x. d/dx(y^2) = d/dy(y^2)* dy/dx = 2y* dy/dx which implies y*y’= d/dx(y^2/2). We may thus write d/dx(y^2/2) = x = d/dx(x^2/2) which implies d/dx(y^2-x^2) = 0. We may thus conclude y(x) = sqrt(x^2+a) which is defined when x^2 > -a.

[@alphastar5626]

( f² )' = 2.f'.f which equals to 2x thanks to the equation we get by derivating the starting statement of the exercise. then you integrate f² ' = 2.x and you find f = ± sqrt( x² + c)

[@kianmath71]

Great video😊

[@امینظاهرزاده]

Well done 👏

[@SyberMath]

Thank you

[@E.h.a.b]

f(x) = x is a possible solution.

[@SyberMath]

That’s right

[@rodrigomarinho1807]

Let c be zero.

[@scottleung9587]

Nice!

[@tarunmnair]

Isnt it a separable DE ? So, we can directly do y.dy = x.dx and then y = sqrt(x^2+c) , I think, u didn't need to use substitution... (2nd method 😂😂😂) Also, even in 2nd substitution, we can substitute u=sin(theta)...

[@user-lu6yg3vk9z]

Try this problem Integral 1/(x^2(x^2+25))dx Hint: Use Fractional Decomposition

[@theblainefarm3310]

I'm curious why the substitution was necessary? I know other people have left the same comment.

[@ChimezieFredAnaekwe]

Multiple strategies, I guess.

[@rakenzarnsworld2]

f(x) = x