query in differential equations.

Question

Seperable differential equations are of the form,

dy/dx = f(x,y).

i.e dy/dx = f(x) g(y).

Suppose in the above equation we have,f(x) as √x and g(y) = y+1. but a book i am studying this topic says y is y(x), we can call y as h(x).

So g(y)=g[h(x)]. right? so what we substitute g[h(x)] in y+1 ? how?

Another thing i didn't get is,

why do we integrate a differential equation after adjusting and showing equality. .i mean we just have to show that rate of change of y=h(x) is equal to the product of f(x) and g(y). what is the reason for integrating.

Answer 1

The answer to your question depends a bit on the type of DE you are asked to solve.

For the example you have, you want to express y(x) as a function of x.

dy/dx=(y+1)√x

dy/(y+1) = sqrt(x) dx

ln (y+1) = (2/3)x^3/2 + C where C is a constant term which depends on the initial conditions.

You can now express y as a function of x although it is a messy relationship.