Question

Show whether the following equation are exact and hence solve the equation.

a) 2x(ye^x² -1)dx + e^x²dy= 0

b) ( 6x⁵y³ + 4x³y⁵) dx + (2x⁶y² - 5x⁴y⁴) dy =0

Answer 1

$a) 2x(ye^{x^2}-1)dx+e^{x^2}dy =0$

$M(x,y)=2x(ye^{x^2}-1)$ and $N(x,y)=e^{x^2}$

$M_y=2xe^{x^2}=N_x$

The given differential equation is exact.

$\int M(x,y)dx=ye^{x^2}-x^2+g(y)$

$\int N(x,y)dy =ye^{x^2}+h(x)$

Answer: $f(x,y)=ye^{x^2}-x^2+C.$

$b) (6x^5y^3+4x^3y^5)dx+(2x^6y^2-5x^4y^4)dy=0$

$M(x,y)=6x^5y^3+4x^3y^5$ and $N(x,y)=2x^6y^2-5x^4y^4$

$M_y=18x^5y^2+20x^3y^4$ and $N_x(x,y)=12x^5y^2-20x^3y^4$

$M_y\neq N_x$

The given differential equation is not exact.