Find the value of k such that the differential equation $\left( x y ^ { 2 } + k x ^ { 5 } y + x ^ { 4 } \right) d x + \left( x ^ { 6 } + x ^ { 2 } y + y ^ { 6 } \right) d y = 0$ is exact.
A) 5
B) 4
C) 8
D) 7
E) 6

Question

Quizplus · Accepted Answer

For a differential equation of the form $M(x, y)dx + N(x, y)dy = 0$ to be exact, the condition $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$ must be satisfied. Here, $M(x, y) = xy^2 + kx^5y + x^4$ and $N(x, y) = x^6 + x^2y + y^6$.Calculating the partial derivatives, we get:$\frac{\partial M}{\partial y} = 2xy + kx^5$ and $\frac{\partial N}{\partial x} = 6x^5 + 2xy$.Setting these equal to each other gives:$2xy + kx^5 = 6x^5 + 2xy$.Solving for $k$, we find that $k = 6$.

Find the Value of K Such That the Differential Equation (xy2+kx5y+x4)dx+(x6+x2y+y6)dy=0\left( x y ^ { 2 } + k x ^ { 5 } y + x ^ { 4 } \right) d x + \left( x ^ { 6 } + x ^ { 2 } y + y ^ { 6 } \right) d y = 0(xy2+kx5y+x4)dx+(x6+x2y+y6)dy=0

Find the Value of K Such That the Differential Equation $\left( x y ^ { 2 } + k x ^ { 5 } y + x ^ { 4 } \right) d x + \left( x ^ { 6 } + x ^ { 2 } y + y ^ { 6 } \right) d y = 0$