= Consider the initial value problem (2xy² D. + cos x cosy (a) Check whether the differential equation is OA. Separable B. Not exact C. O C. OD. cosy) dx + (2x²y- sin x siny) dy = 0 and y(0) = 1 Exact because (2xy² + cos x cos Exact because (2xy² 3 (b) The general solution of the differential equation is given by A. None of the given answers is correct B. 22 x²y² + sinx siny cos x cos y = C with C an arbirary constant 4xy + sin x siny = C with C an arbirary constant 2xy 3 B. + 3 2x y 3 osy), = (2x²y- sinx siny), os y) = (2x²y- sin x siny) y X + cos x COS OC. 2xy + 2x 3 22 OE. x²y²+ sin x cos y = C with C an arbirary constant X 22 x y + sin x cos y = 0 3 3y + sin x sin y cos x cos y = C with C an arbirary constant (c) The solution of the initial value problem is given implicitly by 22 x y + sin x siny- cos x cos y = - cos 1 www + sin x sin y cos x cos y = - cos 1 -

The given initial value problem is ,

  $\left(2x{y}^2+\cos x \cos y\right)dx+\left(2x^{2}y-\sin x \sin y\right)dy=0$

 and   $y\left(0\right) =1$

(.)  A differential equation is said to be           ' separable ' if it can be written in the           form ,

         $F\left(x\right)G\left(y\right) dx + f\left(x\right)g\left(y\right) dy = 0$

(.)  Differential equation ,                                           $M(x,y) dx + N(x,y) dy = 0$

  is said to be exact if ,  

                  $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

  or     ${\left[M(x,y)\right]}_y = {\left[N(x,y)\right]}_x$