Q. 66 - Calculus

Question

Solve the exact differential equations in Exercises 63–66. $\frac{12x^3}{y^5}-\frac{15x^4}{y^6}\frac{dy}{dx}=0.$

Step-by-Step Solution

Answer

The solution of given exact differential equation is: $\frac{3x^4}{y^5}+C=0$

Step 1. Given information

Exact differential equation, $\frac{12x^3}{y^5}-\frac{15x^4}{y^6}\frac{dy}{dx}=0$

Step 2. Solving the given exact differential equation

$$\begin{gathered} \mathrm{Given}, \frac{12x^3}{y^5}-\frac{15x^4}{y^6}\frac{dy}{dx}=0 \\ \Rightarrow \frac{12x^3}{y^5}dx-\frac{15x^4}{y^6}dy=0 \\ \mathrm{It} \mathrm{is} \mathrm{a} \mathrm{exact} \mathrm{differential} \mathrm{equation} \mathrm{of} \mathrm{the} \mathrm{the} \mathrm{form} Mdx+Ndy=0, \\ \mathrm{with} \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}. \mathrm{the} \mathrm{solution} \mathrm{is} \mathrm{given} \mathrm{by}, \\ f(x,y)=\int (\mathrm{treating} y as \mathrm{constant} \mathrm{in} \mathrm{M}) dx+\int (\mathrm{terms} \mathrm{independent} \mathrm{of} x \mathrm{in} N) dy=0 \\ \Rightarrow \int \frac{12x^3}{y^5}dx-\int \left(0\right) dy=0 \\ \Rightarrow \frac{12x^4}{4y^5}+C=0 \\ \Rightarrow \frac{3x^4}{y^5}+C=0 \end{gathered}$$