Q. 65 - Calculus

Question

Solve the exact differential equations in Exercises 63–66. $e^x \ln y+x^3 +\frac{ e^x}{y}\frac{dy}{dx}=0$

Step-by-Step Solution

Answer

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

Step 1. Given information

Exact differential equation, $e^x \ln y+x^3 +\frac{ e^x}{y}\frac{dy}{dx}=0$

Step 2. Solving the given exact differential equation

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