prudencemwaura
Answered

Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

05* Find, for y> 0, the general solution of the differential equation dy/dx=xy.
Inlyl=1/2x^2+c
Inlyl=1/2x^2-c
Inlyl=-1/2x^2-c
Inly|=-1/2x^2+c​

Sagot :

LammettHash

The ODE is separable.

[tex]\dfrac{dy}{dx} = xy \iff \dfrac{dy}y = x\,dx[/tex]

Integrate both sides to get

[tex]\displaystyle \int\frac{dy}y = \int x\,dx[/tex]

[tex]\boxed{\ln|y| = \dfrac12 x^2 + C}[/tex]

But notice that replacing the constant [tex]C[/tex] with [tex]-C[/tex] doesn't affect the solution, since its derivative would recover the same ODE as before.

[tex]\ln|y| = \dfrac12 x^2 - C \implies \dfrac1y \dfrac{dy}{dx} = x \implies \dfrac{dy}{dx} = xy[/tex]

so either of the first two answers are technically correct.

Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.