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# Find the general solution of the following differential equation :  (1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0 - CBSE (Science) Class 12 - Mathematics

ConceptGeneral and Particular Solutions of a Differential Equation

#### Question

Find the general solution of the following differential equation :

(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0

#### Solution

Given:

(1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0

Let tan1y=t

y=tant

=>dy/dx=sec^2tdt/dx

Therefore, the equation becomes

(1+tan2t)+(xet)sec2dt/dx=0

=>sec^2t+(x-e^t)(sec^2t)dt/dx=0

=>1+(x-e^t)dt/dx=0

=>(x-e^t)dt/dx=-1

=>x-e^t=dx/dt

=>dx/dt+1.x=e^t

If =e∫1.dt

= et

:. e^t.(dx/dt+1.x)=e^t.e^t

=>d/dt(xe^t)=e^(2t)

Integrating both the sides, we get

xe^t=inte^(2t)dt

=>xe^t=1/2e^(2t)+C " ....(1)"

Substituting the value of t in (1), we get

xe^(tan^(1))y=1/2e^(2tan^(-1)y)+C_1

=>e^2tan^(-1y)=2xe^(tan^1y)+C

It is the required general solution.

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Solution Find the general solution of the following differential equation :  (1+y^2)+(x-e^(tan^(-1)y))dy/dx= 0 Concept: General and Particular Solutions of a Differential Equation.
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