Q. 64 - Calculus

Question

Solve the exact differential equations in Exercises 63–66. $y \cos \left(xy\right)-3+(x \cos (xy)+2)\frac{dy}{dx}=0.$

Step-by-Step Solution

Answer

The solution of given exact differential equation is: $\sin \left(xy\right)-3x+2y+C=0$

Step 1. Given information

Exact differential equation, $y \cos \left(xy\right)-3+(x \cos (xy)+2)\frac{dy}{dx}=0$

Step 2. Solving the given exact differential equation

$$\begin{gathered} y \cos \left(xy\right)-3+(x \cos (xy)+2)\frac{dy}{dx}=0 \\ \Rightarrow \left(y \cos \left(xy\right)-3\right) dx+(x \cos (xy)+2) dy=0 \\ \mathrm{It} \mathrm{is} \mathrm{a} \mathrm{exact} \mathrm{differential} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{the} \mathrm{form} Mdx+Ndy=0, \\ \mathrm{with} \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}. \mathrm{the} \mathrm{solution} \mathrm{is} \mathrm{given} \mathrm{by}, \\ f(x,y)=\int (\mathrm{treating} y as \mathrm{constant} \mathrm{in} \mathrm{M}) dx+\int (\mathrm{terms} \mathrm{independent} \mathrm{of} x \mathrm{in} N) dy=0 \\ \Rightarrow \int \left(y \cos \left(xy\right)-3\right) dx+\int \left(2\right) dy=0 \\ \Rightarrow \sin \left(xy\right)-3x+2y+C=0 \end{gathered}$$