7. Antiderivatives & Indefinite Integrals

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 $y\left(x\right)$?

Solve the initial value problem: $x^{\prime }=2x$, $x\left(0\right)=4$. What is $x\left(t\right)$?

Solve the initial value problems in Exercises 87 and 88.

87. dy/dx = 1 + 1/x, y(1) = 3

Initial Value Problems

Solve the initial value problems in Exercises 89–92.

dy/dx = (𝓍 + 1/𝓍)² , y(1)= 1

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 $y\left(t\right)$?

A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?

Initial Value Problems

Solve the initial value problems in Exercises 71–90.

d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(v) = sec v tan v; F(0) = 2, -π/2 < v < π/2

Solving initial value problems Find the solution of the following initial value problems.

g'(x) = 7x(x⁶ - 1/7); g(1) = 2

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

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.

v(t) = 2√t; s(0) = 1

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

c. Find the time when the object reaches its highest point. What is the height? 

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

In Exercises 129–132 solve the initial value problem.

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

{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