Question

Question #f5f35

Answer 1

`GS : y=2+ce^{x/6(6+3x-2x^2)}`.

Explanation:

Let us observe that, the given diff. eqn. is,

`dy=(2x^2+y-x^2y+xy-2x-2)dx`

`={(2x^2-2x-2)-(x^2-x-1)y}dx`,

`={2(x^2-x-1)-(x^2-x-1)y}dx`,

`=(x^2-x-1)(2-y)dx`.

`rArr dy/(2-y)=(x^2-x-1)dx, or,`

`dy/(y-2)+(x^2-x-1)dx=0`.

Obviously, it is a separable variable type diff. eqn., and, now,

when the variables are separated, integrating it term by term,

we get its general solution (GS) as shown below :

`intdy/(y-2)+int(x^2-x-1)dx=lnc`.

`:. ln(y-2)+(x^3/3-x^2/2-x)=lnc`.

`:. ln(y-2)+x/6(2x^2-3x-6)=lnc`.

`:. ln((y-2)/c)=x/6(6+3x-2x^2)`.

`:. (y-2)/c=e^{x/6(6+3x-2x^2)}`.

`rArr y=2+ce^{x/6(6+3x-2x^2)},` is the desired GS.

Enjoy Maths., and Spread the Joy!