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PLEASE SOLVE “ dy/dx=sin(x+y)+cos(x+y) “ and please tell me the genaral solution of the first degree differential equation of given

PLEASE SOLVE “ dy/dx=sin(x+y)+cos(x+y) “
  and please tell me the genaral solution of the first degree differential equation of given

Grade:12

1 Answers

BALAJI ANDALAMALA
askIITians Faculty 78 Points
6 years ago
Put x+y = v
1+dy/dx = dv/dx
dy/dx = dv/dx – 1
\frac{dy}{dx} = sin(x+y)+cos(x+y)
\frac{dv}{dx}-1= sin(v)+cos(v)
Using variable and separable method
\frac{dv}{1+cosv+sinv}= dx
Integrate both sides
\int \frac{dv}{1+cosv+sinv}= \int dx
\int \frac{dv}{1+\frac{1-tan^2\frac{v}{2}}{1+tan^2\frac{v}{2}}+\frac{2tan\frac{v}{2}}{1+tan^2\frac{v}{2}}}= \int dx
\int \frac{sec^2\frac{v}{2}dv}{2(1+tan\frac{v}{2})}= \int dx
ln(1+tan\frac{v}{2})= x+c
ln(1+tan\frac{x+y}{2})= x+c
This is the solution.

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