Q. 22

# Solve the differe

Given,

(tan-1 y – x)dy = (1 + y2)dx

Clearly, this is a linear differential equation. Comparing with the standard form

A solution of such equation is given by:

x(I.F) = where I.F = integrating factor

We get P(y) = & Q(y) =

Integrating factor I.F is given :

I.F =

We know that:

I.F =

The solution is given as:

…(1)

Where I =

Let tan-1y = u

du =

I =

Using integration by parts:

I =

I =

I =

the solution is given using equation 1:

OR

Given,

…(1)

Clearly, the equation is homogeneous(can be observed directly)

We know that for solving a homogeneous we differential equation,

We take y = vx.

As y = vx

Differentiating both sides w.r.t x we get

equation 1 can be rewritten as-

Integrating both sides we get-

The above equation gives the general solution. For a particular solution, we need to find the value of C.

As given that at x = 0 ,y = 1

C = 0.

Particular solution at x = 0 and x =1 is given by:

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