Question

What is the initial condition in this set of code?

Answer 1

Answer:An initial condition is an extra bit of information about a differential equation that tells you the value of the function at a particular point. Differential equations with initial conditions are commonly called initial value problems.

The video above uses the example

{

d

y

d

x

=

cos

(

x

)

y

(

0

)

=

−

1

to illustrate a simple initial value problem. Solving the differential equation without the initial condition gives you  

y

=

sin

(

x

)

+

C

.

Once you get the general solution, you can use the initial value to find a particular solution which satisfies the problem. In this case, plugging in  

0

for  

x

and  

−

1

for  

y

gives us  

−

1

=

C

, meaning that the particular solution must be  

y

=

sin

(

x

)

−

1

.

So the general way to solve initial value problems is: - First, find the general solution while ignoring the initial condition. - Then, use the initial condition to plug in values and find a particular solution.

Two additional things to keep in mind: First, the initial value doesn't necessarily have to just be  

y

-values. Higher-order equations might have an initial value for both  

y

and  

y

′

, for example.

Second, an initial value problem doesn't always have a unique solution. It's possible for an initial value problem to have multiple solutions, or even no solution at all.

Explanation: