7. Antiderivatives & Indefinite Integrals

Solve the initial value problems in Exercises 53–56 for y as a function of x.

(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1

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

Express the solutions of the initial value problems in Exercises 35 and 36 in terms of integrals.

dy/dx = sin x/x , y(5) = -3

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

Solve the initial value problems in Exercises 115–120.

115. dy/dx = 1/√(1 - x²), y(0) = 0

Solve the initial value problems in Exercises 115–120.

117. dy/dx = 1/(x√(x² - 1)), x > 1; y(2) = π