Textbook Question
33–42. Solving initial value problems Solve the following initial value problems.
y'(x) = 4 sec² 2x, y(0) = 8
38
views
13. Intro to Differential Equations
Basics of Differential Equations
1:59 minutesProblem 9.1.21
Textbook Question
21–32. Finding general solutions Find the general solution of each differential equation. Use C,C1,C2... to denote arbitrary constants.
y'(t) = 3 + e⁻²ᵗ
Verified step by step guidance1
Recognize that the given differential equation is a first-order ordinary differential equation of the form \(y'(t) = f(t)\), where \(f(t) = 3 + e^{-2t}\).
To find the general solution, integrate both sides with respect to \(t\): \(\int y'(t) \, dt = \int (3 + e^{-2t}) \, dt\).
Split the integral on the right side into two separate integrals: \(\int 3 \, dt + \int e^{-2t} \, dt\).
Integrate each term: the integral of \$3\( with respect to \)t\( is \)3t\(, and the integral of \)e^{-2t}\( with respect to \)t$ is \(-\frac{1}{2} e^{-2t}\) (using substitution or recognizing the exponential integral).
Combine the results and add the arbitrary constant of integration \(C\) to write the general solution: \(y(t) = 3t - \frac{1}{2} e^{-2t} + C\).
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
General Solution of a Differential Equation
The general solution of a differential equation includes all possible solutions and typically contains arbitrary constants. It represents the family of functions that satisfy the equation, capturing both particular and homogeneous parts.
Recommended video:
04:00
Solutions to Basic Differential Equations
Integration of Functions
Solving first-order differential equations often involves integrating the right-hand side with respect to the independent variable. Integration reverses differentiation and helps find the original function from its derivative.
Recommended video:
05:11Integrals of General Exponential Functions
Exponential Functions and Their Integrals
Exponential functions like e^(-2t) have specific integral formulas. For example, ∫e^(kt) dt = (1/k)e^(kt) + C, where k is a constant. Recognizing and integrating these correctly is essential for solving differential equations involving exponentials.
Recommended video:
05:11Integrals of General Exponential Functions
Related Videos
Related Practice