Back{Use of Tech} Graphing general solutions Graph several functions that satisfy each of the following differential equations. Then find and graph the particular function that satisfies the given initial condition.
f'(x) = 3x + sinx; f(0) = 3
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dr/dθ = −π sin (πθ), r(0) = 0
104–107. Functions from derivatives Find the function f with the following properties.
ƒ'(t) = sin t + 2t; ƒ(0) = 5
Solve the initial value problems in Exercises 55–58.
55. dy/dt = e^t sin(e^t − 2),y(ln 2) = 0
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
ds/dt = 1 + cos t, s(0) = 4
Solve the initial value problems in Exercises 115–120.
115. dy/dx = 1/√(1 - x²), y(0) = 0
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = -32; v(0) = 20, s(0) = 0
Solve the initial value problems in Exercises 87 and 88.
88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
dy/dx = 3x⁻²ᐟ³, y(−1) = −5
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:
;
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = 2 + 3 sin t; v(0) = 1, s(0) = 10
Find the function that satisfies the following differential equation.
; ;
Initial Value Problems
Find the curve y = f(x) in the xy-plane that passes through the point (9,4) and whose slope at each point is 3√x.