Question

In Exercises 1–22, solve the differential equation.(1+eˣ) dy + (yeˣ + e⁻ˣ) dx = 0

Accepted Answer

Rewrite the given differential equation in the form $$M(x,y)\,dx + N(x,y)\,dy = 0. $$Here, identify $$M(x,y)$$ and $$N(x,y)$$ from the equation $$(1+e^{x}$$)\,dy + $$(ye^{x}$$ + $$e^{-x}$$)\,dx = 0$$. This gives M(x$$,y) = $$ye^{x}$$ + $$e^{-x}$$ and $$N(x,y) = 1 + e$$^{x}$$. Check if the differential equation is exact by computing the partial derivatives $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}$$. Calculate $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}$$ and compare them to see if they are equal. If the equation is not exact, find an integrating factor that depends on either x$ or y$ to make it exact. To do this, use the formula for integrating factors: for example, if $$\frac{\partial}{\partial y} \left( \frac{M}{N} \right)$$ depends only on x$, then an integrating factor depending on x$ might exist, and similarly for y$. Once the equation is exact (either originally or after multiplying by the integrating factor), find the potential function $$\Psi(x,y)$$ such that $$\frac{\partial \Psi}{\partial x} = M$$ and $$\frac{\partial \Psi}{\partial y} = N$$. Integrate M$ with respect to x$ to find $$\Psi(x,y)$$ up to a function of y$, then differentiate this expression with respect to y$ and set it equal to N$ to determine the unknown function of y$. Write the implicit solution as $$\Psi(x,y) = C$$, where C$ is a constant. This represents the general solution to the differential equation.

In Exercises 1–22, solve the differential equation.

(1+eˣ) dy + (yeˣ + e⁻ˣ) dx = 0

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Differential Equations and Their Forms

Exact Differential Equations

Integrating Factors