Question

Write an equivalent first-order differential equationand initial condition for yy= 1+int_0^x y(t) dt

First order differential equations
ANSWERED
asked 2020-11-23

Write an equivalent first-order differential equationand initial condition for y \(y= 1+\int_0^x y(t) dt\)

Expert Answers (1)

2020-11-24

\(y= 1+\int_0^x y(t) dt\) (1)
First we find the first-order differential equation by differentiating both sides with respect to x
\(dy/dx= d/dx(1+\int_{0}^{x} y(t)dt)\) (2)
As according to fundamental theorem of calculus
\(d/dx \int_{a}^{x} f(t)dt= f(x)\)
So we can write (2) as
\(dy/dx= y(x)\ or\ dy/dx=y\), \(y'=y\)
Now we find the initial conditions for y
The equation is of the form
\(y= f(c)+\int_{c}^{x} g(t)dt\)
Now on comparing this with equation (1) we get
\(f(0)=1\ and\ c=0\)
So, the initial condition for y is \(y(0)=1\)
Therefore the first order differential equation is \(y'=y\) and initial condition for y is \(y(0)=1\)

49
 
Best answer

expert advice

Have a similar question?
We can deal with it in 3 hours
...