Answer:
[tex]y(x)=4a\sqrt{3}* sin(x)-3a[/tex]
Step-by-step explanation:
We have a separable equation, first let's rewrite the equation as:
[tex]\frac{dy(x)}{dx} =\frac{3a+y}{tan(x)}[/tex]
But:
[tex]\frac{1}{tan(x)} =cot(x)[/tex]
So:
[tex]\frac{dy(x)}{dx} =cot(x)*(3a+y)[/tex]
Multiplying both sides by dx and dividing both sides by 3a+y:
[tex]\frac{dy}{3a+y} =cot(x)dx[/tex]
Integrating both sides:
[tex]\int\ \frac{dy}{3a+y} =\int\cot(x) \, dx[/tex]
Evaluating the integrals:
[tex]log(3a+y)=log(sin(x))+C_1[/tex]
Where C1 is an arbitrary constant.
Solving for y:
[tex]y(x)=-3a+e^{C_1} sin(x)[/tex]
[tex]e^{C_1} =constant[/tex]
So:
[tex]y(x)=C_1*sin(x)-3a[/tex]
Finally, let's evaluate the initial condition in order to find C1:
[tex]y(\frac{\pi}{3} )=3a=C_1*sin(\frac{\pi}{3})-3a\\ 3a=C_1*\frac{\sqrt{3} }{2} -3a[/tex]
Solving for C1:
[tex]C_1=4a\sqrt{3}[/tex]
Therefore:
[tex]y(x)=4a\sqrt{3}* sin(x)-3a[/tex]
The solution of the differential equation that satisfies the given initial condition is [tex]y = a\cdot (0.817\cdot 10^{\sin x}-3)[/tex].
In this question we must separate variables on each side of the formula:
[tex]\frac{dy}{dx} \cdot \tan x = 3\cdot a + y[/tex]
[tex]\int {\frac{dx}{\tan x} } = \int {\frac{dy}{3\cdot a+y} }[/tex]
[tex]\log (\sin x) + C = \log(3\cdot a + y)[/tex]
[tex]C\cdot 10^{\sin x} = 3\cdot a + y[/tex]
[tex]C = \frac{3\cdot a + y}{10^{\sin x}}[/tex] (1)
Where [tex]C[/tex] is the integration constant. If we know that [tex]y\left(\frac{\pi}{3} \right) = 3\cdot a[/tex], then the integration constant is:
[tex]C = \frac{6\cdot a}{10^{\frac{\sqrt{3}}{2} }}[/tex]
[tex]C \approx 0.817\cdot a[/tex] (2)
By (2) in (1):
[tex]0.817\cdot a \cdot 10^{\sin x} = 3\cdot a + y[/tex]
[tex]y = a\cdot (0.817\cdot 10^{\sin x}-3)[/tex] (3)
The solution of the differential equation that satisfies the given initial condition is [tex]y = a\cdot (0.817\cdot 10^{\sin x}-3)[/tex]. [tex]\blacksquare[/tex]
To learn more on differential equations, we kindly invite to check this verified question: https://brainly.com/question/25731911
