- #1
karush
Gold Member
MHB
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it's late so I'll just start this
\[ \dfrac{dy}{dx}=\dfrac{2\cos 2x}{3+2y} \]
so \[(3+2y) \, dy= (2\cos 2x) \, dx\]
$y^2 + 3 y= sin(2 x) + c$
Last edited:
(-1)^2 + 3 (-1)= sin(2 (0)) + c 1-3=1+c -3=c |
karush said:\(\displaystyle y(x)=\dfrac{-3+\sqrt{9+4(\sin(2x)-2)}}{2}
=\dfrac{-3+\sqrt{4\sin(2x)+1}}{2} =-\dfrac{3}{2}+\sqrt{\sin(2x)+\dfrac{1}{4}}\)
why is $x=\dfrac{\pi}{4}$
ok I see. $2(\pi/4)=\pi/2$skeeter said:what value of x makes sin(2x) a maximum?
A differential equation is a mathematical equation that involves an unknown function and its derivatives. It describes how the value of a function changes in relation to its input variables.
To solve a differential equation, you need to find the function that satisfies the equation. This can be done by using various methods such as separation of variables, integrating factors, or using specific formulas for certain types of equations.
The general solution to the differential equation dy/dx = (2cos 2x)/(3+2y) is y = -3/2 + 1/2tan(2x + C), where C is a constant of integration.
To find the particular solution, you need to use initial conditions. Plug in the values of the initial conditions into the general solution and solve for the constant of integration, C. This will give you the specific function that satisfies the given differential equation.
Differential equations are used to model and understand various phenomena in science, such as population growth, chemical reactions, and motion of objects. By solving these equations, scientists can make predictions and gain a deeper understanding of the natural world.