Question

Determine whether each of the equations is exact. If it is exact, find the solution.

1. (dy/dx)=-(ax-by/bx-cy)
2. (2x+4y)+(2x-2y)y'=0
3. (3x²-2xy+2)+(6y²-x²+3)y'=0
4. (ycosx+2xe^y)+(sinx+x²e^y-1)y'=0

Answer 1

$\text{Differential equation,}\\ M(x,y)dx+N(x,y)=0\\ \text{then, if partial derivative of M w.r.t y is equal to the partial derivative of N w.r.t x. ---(1)}\\ 1.\\ \frac{dy}{dx}=-\frac{ax-by}{bx-cy}\\ (ax-by)dx+(bx-cy)dy=0\\ Now,\\ M_y=-b\\ N_x=b\\ \text{This is not exact.}\\ 2.\\ (2x+4y)dx+(2x-2y)dy=0\\ Now,\\ M_y=4\\ N_x=2\\ \text{This is not exact.}\\ 3.\\ (3x^2-2xy+2)dx+(6y^2-x^2+3)dy=0\\ Now,\\ M_y=-2x\\ N_x=-2x\\ \text{This is an exact differential equation.}\\ \text{Then the solution is given by,}\\ \int(3x^2-2xy+2)dx+\int(6y^2-x^2+3)dy=c\\ x^3-x^2y+2x+2y^3+3y=c\\ 4.\\ (ycosx+2xe^y)dx+(sinx+x^2e^y-1)dy=0 Now,\\ M_y=cosx+2xe^y\\ N_x=cosx+2xe^y\\ \text{This is an exact differential equation.}\\ \text{Then the solution is given by,}\\ \int(ycosx+2xe^y)dx+\int(sinx+x^2e^y-1)dy=c\\ ysinx+x^2e^y-y=c$