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In Problems 1–22, is the solution of the given differential equation by separation of variables. You may obtain a detailed answer by clicking on the question.

1: \frac{dy}{dx} = \sin(5x) Step by Step Solution

Here we Solve by separation of variables dy/dx= sin5x. .

Given differential equation is:

(1)   \begin{equation*} \frac{dy}{dx} = \sin(5x) \end{equation*}

We can separate the variables as follows:

(2)   \begin{equation*} dy = \sin(5x)dx \end{equation*}

Now, we can integrate both sides of the equation:

(3)   \begin{equation*} \int dy = \int \sin(5x)dx \end{equation*}

Integrating the left-hand side, we get:

(4)   \begin{equation*} y = \int dy = y + C_1 \end{equation*}

where C_1 is the constant of integration.

For the right-hand side, we can use the substitution u = 5x and du/dx = 5 to obtain:

(5)   \begin{equation*} \int \sin(5x)dx = \frac{1}{5}\int \sin(u)du = -\frac{1}{5}\cos(u) + C_2 \end{equation*}

where C_2 is another constant of integration. Substituting back u=5x, we have:

(6)   \begin{equation*} \int \sin(5x)dx = -\frac{1}{5}\cos(5x) + C_2 \end{equation*}

Substituting this back into our original equation, we have:

(7)   \begin{equation*} y = -\frac{1}{5}\cos(5x) + C_2 + C_1 \end{equation*}

where C_1 and C_2 are constants of integration.

Therefore, the solution to the differential equation \frac{dy}{dx} = \sin(5x) is:

(8)   \begin{equation*} y = -\frac{1}{5}\cos(5x) + C \end{equation*}

where C = C_1 + C_2 is the constant of integration.

1. \frac{dy}{dx} = \sin(5x)

2. \frac{dy}{dx} = (x+1)^2

3. dx + e^{3x}dy = 0

4. dy - (y-1)^2 dx = 0

5. x \frac{dy}{y} = 4y

6. \frac{dy}{dx} = 2xy^2

7. \frac{dy}{dx} = e^{3x+2y}

8. } e^x y\frac{dy}{dx} = e^{-y} + e^{-2x-y}

9. y \ln x \frac{dx}{dy} = \left(\frac{y+1}{x}\right)^2

10. } \frac{dy}{dx} = \left(\frac{2y+3}{4x+5}\right)^2

11. \cosec(y)dx+\sec^2(x)dy=0

12. \sin(3x)dx + 2y\cos^3(3x)dy = 0

13. (e^y+1)^2e^{-y}dx+(e^x+1)^3e^{-x}dy=0

14. x(1+y^2)^{\frac{1}{2}}dx=y(1+x^2)^{\frac{1}{2}}dy

15. \frac{dS}{dr}=kS

16. \frac{dQ}{dt} = k(Q-70)

17. \frac{dP}{dt} = P-P^2

18. \frac{dN}{dt} + N = Nte^{t+2}

19. \frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}

20. \frac{dy}{dx}=\frac{xy+2y-x-2}{xy-3y+x-3}

21. \frac{dy}{dx}=x\sqrt{1-y^2}

22. \left(e^x+e^{-x}\right)\frac{dy}{dx}=y^2

  1. Separable Differential Equations Solutions
  2. First Order Differential Equations

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