Write an equivalent first-order differential equationand initial condition for y y=1+∫0xy(t)dt

naivlingr

naivlingr

Answered

2020-11-23

Write an equivalent first-order differential equationand initial condition for y y=1+0xy(t)dt

Answer & Explanation

liannemdh

liannemdh

Expert

2020-11-24Added 106 answers

y=1+0xy(t)dt (1)
First we find the first-order differential equation by differentiating both sides with respect to x
dy/dx=d/dx(1+0xy(t)dt) (2)
As according to fundamental theorem of calculus
d/dxaxf(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)+cxg(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

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