Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

7. Antiderivatives & Indefinite Integrals

Initial Value Problems

Calculus

7. Antiderivatives & Indefinite Integrals

Initial Value Problems

Previous problemNext problem

3:23 minutes

Problem 4.9.85

Textbook Question

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

y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ); y (π/4) = 3, -π/2 < Θ < π/2

Verified step by step guidance

Step 1: Recognize that this is an initial value problem involving a first-order differential equation. The goal is to find the function y(Θ) that satisfies both the differential equation y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ) and the initial condition y(π/4) = 3.

Step 2: Begin by integrating the given derivative y'(Θ) to find y(Θ). Rewrite the differential equation as y'(Θ) = (√2 cos³ Θ / cos² Θ) + (1 / cos² Θ). Simplify the terms: y'(Θ) = √2 cos Θ + sec² Θ.

Step 3: Break the integral into two parts: ∫(√2 cos Θ) dΘ + ∫(sec² Θ) dΘ. Use standard integration rules. For the first term, ∫(√2 cos Θ) dΘ integrates to √2 sin Θ. For the second term, ∫(sec² Θ) dΘ integrates to tan Θ.

Step 4: Combine the results of the integration to express y(Θ): y(Θ) = √2 sin Θ + tan Θ + C, where C is the constant of integration.

Step 5: Use the initial condition y(π/4) = 3 to solve for the constant C. Substitute Θ = π/4 into the equation: y(π/4) = √2 sin(π/4) + tan(π/4) + C. Simplify using trigonometric values (sin(π/4) = √2/2 and tan(π/4) = 1) to find C. Once C is determined, substitute it back into the equation to finalize the solution for y(Θ).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Initial Value Problem (IVP)

An initial value problem is a type of differential equation that specifies the value of the unknown function at a particular point. In this context, it involves finding a function y(Θ) that satisfies a given differential equation and meets a specified condition, y(π/4) = 3. Solving an IVP typically requires integrating the differential equation and applying the initial condition to determine any constants of integration.

Differential Equations

A differential equation is an equation that relates a function to its derivatives. In this case, the equation y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ) describes how the function y changes with respect to the variable Θ. Understanding how to manipulate and solve differential equations is crucial for finding the function y(Θ) that satisfies the given relationship.

Integration Techniques

Integration techniques are methods used to find the integral of a function, which is essential for solving differential equations. In this problem, one may need to apply techniques such as substitution or integration by parts to evaluate the integral of the right-hand side of the differential equation. Mastery of these techniques allows for the determination of the function y(Θ) from its derivative.

Watch next

Master Initial Value Problems with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

Textbook Question

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.

views

Textbook Question

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) = 2t + 4; s(0) = 0

views

Textbook Question

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

views

Textbook Question

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

g'(x) = 7x(x⁶ - 1/7); g(1) = 2

views

Textbook Question

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) = 6t² + 4t - 10; s(0) = 0

views

Textbook Question

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

views

Textbook Question

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

views

Textbook Question

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?

views

Show more practices