Solve the following differential equation algebraically
dy/dx = xy^2 + 4x y(0) = 0
I reached the point dy/(y^2+4) = xdx but I don't know how to solve the left side of the equation.
sorry I responded too early, thank you so much
dy/dx = xy^2 + 4x
dy/dx = x(y^2+4)
dy/(y^2+4) = x dx
now just integrate both sides.
1/2 arctan(y/2) = 1/2 x^2 + c/2
y/2 = tan(x^2 + c)
y = 2 tan(x^2+c)
I got that, but I wasn't sure how to integrate dy/(x^2+4)
To solve the differential equation dy/dx = xy^2 + 4x algebraically, you correctly rearranged the equation to obtain dy/(y^2+4) = xdx. Now, let's integrate both sides of the equation with respect to their respective variables.
Integrating the left side ∫dy/(y^2+4) yields:
∫dy/(y^2+4) = ∫dx
Now, let's solve these integrals step by step. The integral on the left side can be rewritten using a trigonometric substitution. Consider letting y = 2tan(θ), which implies dy = 2sec^2(θ) dθ.
Replacing y and dy in the integral, we have:
∫(2sec^2(θ))/(4tan^2(θ) + 4) dθ = ∫(2sec^2(θ))/(4(tan^2(θ) + 1)) dθ
Simplifying the denominator:
∫(2sec^2(θ))/(4sec^2(θ)) dθ = ∫1/2 dθ
Now, solving this integral yields:
(1/2)θ + C1
where C1 is the constant of integration.
On the right side, we have:
∫dx = x + C2
where C2 is another constant of integration.
Now, equating the two results obtained, we have:
(1/2)θ + C1 = x + C2
To proceed further, it is necessary to compute the inverse tangent function of both sides. This step will allow us to express θ and the solution, y, in terms of x.
θ = 2arctan(y/2)
Substituting this expression into the equation, we get:
(1/2)(2arctan(y/2)) + C1 = x + C2
arctan(y/2) + C1 = x + C2
Now, we can solve for y:
arctan(y/2) = x + C3
where C3 = C2 - C1.
Finally, taking the tangent of both sides, we have:
y/2 = tan(x + C3)
Multiplying both sides by 2, we obtain the solution to the differential equation:
y = 2tan(x + C3)
To find the particular solution for the initial condition y(0) = 0, substitute x = 0 and y = 0 into the solution:
0 = 2tan(0 + C3)
Since the tangent of an angle is zero when the angle is a multiple of π, we have:
0 + C3 = nπ
where n is an integer.
Therefore, the particular solution for the initial condition is:
y = 2tan(x + nπ)
Please note that there are infinitely many solutions to the given differential equation, as indicated by the constant n in the solution.