1. **Problem statement:** Find the functions $y$ given their derivatives: 3. $\frac{dy}{dx} = \frac{2}{x^2 + 4}$ 4. $\frac{dy}{dx} = 3e^x$, with initial condition $y(0) = 4$ 5. $\frac{dy}{dx} = 4y$, with initial condition $y(0) = 3$ --- 2. **Formulas and rules:** - To find $y$ from $\frac{dy}{dx}$, integrate the right side with respect to $x$. - For initial conditions, use $y(x_0) = y_0$ to find the constant of integration. - For differential equations like $\frac{dy}{dx} = ky$, the solution is $y = Ce^{kx}$. --- 3. **Solution for problem 3:** $$y = \int \frac{2}{x^2 + 4} dx$$ Rewrite denominator: $x^2 + 4 = x^2 + 2^2$ Use formula $\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$ So, $$y = 2 \int \frac{dx}{x^2 + 2^2} = 2 \cdot \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C = \arctan\left(\frac{x}{2}\right) + C$$ --- 4. **Solution for problem 4:** Given $\frac{dy}{dx} = 3e^x$, integrate: $$y = \int 3e^x dx = 3 \int e^x dx = 3e^x + C$$ Use initial condition $y(0) = 4$: $$4 = 3e^0 + C = 3 + C \implies C = 1$$ Final solution: $$y = 3e^x + 1$$ --- 5. **Solution for problem 5:** Given $\frac{dy}{dx} = 4y$, this is a separable differential equation. Rewrite: $$\frac{dy}{y} = 4 dx$$ Integrate both sides: $$\int \frac{1}{y} dy = \int 4 dx$$ $$\ln|y| = 4x + C$$ Exponentiate both sides: $$|y| = e^{4x + C} = e^C e^{4x}$$ Let $A = e^C$, so $$y = A e^{4x}$$ Use initial condition $y(0) = 3$: $$3 = A e^{0} = A \implies A = 3$$ Final solution: $$y = 3 e^{4x}$$