Consider the initial value problem: $y^{\mathrm{\text{'}\text{'}}}+16y=\cos \left(4t\right)$, with $y\left(0\right)=0$ and $y^{\mathrm{\text{'}}}\left(0\right)=0$. What is the particular solution $y\left(t\right)$?

Solve the initial value problems in Exercises 71–90.

y⁽⁴⁾ = −sin t + cos t;

y′′′(0) =7, y′′(0) = y′(0) = −1, y(0) = 0

Solve the following initial value problem:

$\frac{dy}{dx}=2x-5$; $y\left(0\right)=4$

Using the acceleration function below, find the velocity function, if the velocity is v = 5 at time t = 2.

$a\left(t\right)=-20$

Find the function $f\left(x\right)$ that satisfies the following differential equation.

$f^{\prime\prime}\left(x\right)=3x^2$; $f^{\prime}\left(0\right)=1$; $f\left(1\right)=3$

Solve the following initial-value problem using Laplace transforms: $y^{\text{'}\text{'}}+4y=0$, $y\left(0\right)=2$, $y^{\text{'}}\left(0\right)=0$. What is $y\left(t\right)$?

Solve the initial value problem: The differential equation is homogeneous. $x{y}^2\frac{dy}{dx}=y^3-x^3$, $y\left(1\right)=4$. What is the explicit solution $y\left(x\right)$?

Solve the initial-value problem: $x(x+1)\frac{dy}{dx}+xy=1$, $y\left(e\right)=1$. What is the solution $y\left(x\right)$?

Solve the initial value problem for the homogeneous differential equation $(x^2+2y^2)\frac{dx}{dy}=xy$, with the initial condition $y(-1)=2$. What is the explicit solution $y\left(x\right)$?