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## Differential Equations : Initial-Value Problems

The initial value is six in mathematical terms is,

Then taking the anti-derivative, you include a C value:

Then, using the initial condition given, we can solve for the value of C:

## Example Question #3 : Initial Value Problems

## Example Question #4 : Initial Value Problems

So this is a separable differential equation with a given initial value.

To start off, gather all of the like variables on separate sides.

Then integrate, and make sure to add a constant at the end

Plug in the initial condition to get:

Solve the separable differential equation

To start off, gather all of the like variables on separate sides.

Notice that when you divide sec(y) to the other side, it will just be cos(y),

and the csc(x) on the bottom is equal to sin(x) on the top.

In order to solve for y, we just need to take the arcsin of both sides:

Solve the differential equation

Then, after the anti-derivative, make sure to add the constant C:

Taking the anti-derivative once, we get:

## Example Question #8 : Initial Value Problems

Solve the differential equation for y

subject to the initial condition:

Then taking the square root to solve for y, we get:

## Example Question #2 : Initial Value Problems

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## no of solutions of the initial value problem?

$x \dfrac{dy}{dx} = y , y (0) = 0, x \geq 0 .$

$\dfrac{dy}{y} = \dfrac{dx}{x},$ by variable separable method, we get

so $y = x -1 $is the required solution.

But it's given that it has uncountable number of solution. I don't understand it.

## 2 Answers 2

The given initial condition is not sufficient to solve for the unknown constant, because:

$$0=\text{C}\cdot0\Longleftrightarrow \text{C}=\frac{0}{0}$$

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## Module 4: Differential Equations

Initial-value problems, learning outcomes.

- Identify an initial-value problem
- Identify whether a given function is a solution to a differential equation or an initial-value problem

## Example: Verifying a Solution to an Initial-Value Problem

You can view the transcript for this segmented clip of “4.1.5” here (opens in new window) .

Verify that [latex]y=3{e}^{2t}+4\sin{t}[/latex] is a solution to the initial-value problem

First verify that [latex]y[/latex] solves the differential equation. Then check the initial value.

## Example: Solving an Initial-value Problem

Solve the following initial-value problem:

Solve the initial-value problem

[latex]y=\frac{1}{3}{x}^{3}-2{x}^{2}+3x - 6{e}^{x}+14[/latex]

Figure 4. Possible velocities for the rising/falling baseball.

Notice that this differential equation remains the same regardless of the mass of the object.

## Example: Velocity of a Moving Baseball

- Find the velocity [latex]v\left(t\right)[/latex] of the baseball at time [latex]t[/latex].
- What is its velocity after [latex]2[/latex] seconds?

[latex]v\left(t\right)=-9.8t[/latex]

If the velocity function is known, then it is possible to solve for the position function as well.

## Example: Height of a Moving Baseball

- Find the position [latex]s\left(t\right)[/latex] of the baseball at time [latex]t[/latex].
- What is its height after [latex]2[/latex] seconds?

Next we substitute [latex]t=0[/latex] and solve for [latex]C\text{:}[/latex]

- 4.1.5. Authored by : Ryan Melton. License : CC BY: Attribution
- Calculus Volume 2. Authored by : Gilbert Strang, Edwin (Jed) Herman. Provided by : OpenStax. Located at : https://openstax.org/books/calculus-volume-2/pages/1-introduction . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike . License Terms : Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction

## How to solve initial value problems

## What is an initial value problem?

I create online courses to help you rock your math class. Read more.

## How to solve initial value problems by using the initial condition

## Take the course

Given ???f'(x)=2??? and ???f(0)=-3???, find ???f(x)???.

Notice that the solution would have been different had we been given a different initial condition.

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## IMAGES

## VIDEO

## COMMENTS

This calculus video tutorial explains how to solve the initial value problem as it relates to separable differential equations.

https://www.patreon.com/ProfessorLeonardExploring Initial Value problems in Differential Equations and what they represent.

Initial Value Problems : Example Question #2 ... Explanation: So this is a separable differential equation, but it is also subject to an initial condition. This

xdydx=y,y(0)=0,x≥0.

In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value

Usually a given differential equation has an infinite number of solutions, so it is natural to ask which one we want to use. To choose one solution

What is an initial value problem? · How to solve initial value problems by using the initial condition · Take the course · Find the antiderivative

where c and k0, k1, ..., kn−1 are given numbers. ... (1) Does a given initial-value problem have a solution? That is, do solutions to the problem exist?

Using the initial data, plug it into the general solution and solve for c. EXAMPLE 1: Solve the initial value problem. SOLUTION:.

Initial value problems in calculus concern differential equations with a known initial condition that specifies the value of the function at