Determine whether the given differential equation is exact, If it is exact, solve it. (2ysin x cos x - y+ 2y²e*»° ) dx = (x - sin x - 4xye) dy Upload Choose a File
Determine whether the given differential equation is exact, If it is exact, solve it. (2ysin x cos x - y+ 2y²e*»° ) dx = (x - sin x - 4xye) dy Upload Choose a File
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Determine whether the given differential equation is exact. If it is exact, solve it.
\[
(2y \sin x \cos x - y + 2y^2 e^{xy^2}) \, dx = (x - \sin^2 x - 4xy e^{xy^2}) \, dy
\]
**Explanation:**
To determine if the differential equation is exact, check if the following condition is satisfied:
Given the differential equation in the form:
\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]
where \(M(x, y) = 2y \sin x \cos x - y + 2y^2 e^{xy^2}\) and \(N(x, y) = x - \sin^2 x - 4xy e^{xy^2}\).
The equation is exact if:
\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]
If this condition holds, solve the equation by finding a function \(\Psi(x, y)\) such that:
\[
\frac{\partial \Psi}{\partial x} = M(x, y) \quad \text{and} \quad \frac{\partial \Psi}{\partial y} = N(x, y)
\]
Integrate \(M(x, y)\) with respect to \(x\) and \(N(x, y)\) with respect to \(y\) to find \(\Psi(x, y)\). Then, solve for the constant of integration.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F74a43f44-8ce4-41e9-89f6-5a77f02dff59%2Fea96921e-be0d-4614-856e-a42e200fe71f%2Fa0rd71g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine whether the given differential equation is exact. If it is exact, solve it.
\[
(2y \sin x \cos x - y + 2y^2 e^{xy^2}) \, dx = (x - \sin^2 x - 4xy e^{xy^2}) \, dy
\]
**Explanation:**
To determine if the differential equation is exact, check if the following condition is satisfied:
Given the differential equation in the form:
\[
M(x, y) \, dx + N(x, y) \, dy = 0
\]
where \(M(x, y) = 2y \sin x \cos x - y + 2y^2 e^{xy^2}\) and \(N(x, y) = x - \sin^2 x - 4xy e^{xy^2}\).
The equation is exact if:
\[
\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}
\]
If this condition holds, solve the equation by finding a function \(\Psi(x, y)\) such that:
\[
\frac{\partial \Psi}{\partial x} = M(x, y) \quad \text{and} \quad \frac{\partial \Psi}{\partial y} = N(x, y)
\]
Integrate \(M(x, y)\) with respect to \(x\) and \(N(x, y)\) with respect to \(y\) to find \(\Psi(x, y)\). Then, solve for the constant of integration.
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