In Exercises 5–8, show that each function is a solution of the given initial value problem.
7. Differential Equation: xy' + y = -sin(x), x>0
Initial condition: y(π/2) = 0
Solution candidate: y = cos(x)/x

In Exercises 129–132 solve the initial value problem.

129. dy/dx = e^(-x-y-2), y(0) = -2

In Exercises 129–132 solve the initial value problem.

131. x dy - (y + √y)dx = 0, y(1) = 1

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0

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$

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 $y\left(t\right)$?