Find the solution of the differential equation that satisfies the given initial

facas9 2021-10-18 Answered

Find the solution of the differential equation that satisfies the given initial condition. \(dy/dx=x/y, y(0)=-3\)

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Nathaniel Kramer
Answered 2021-10-19 Author has 12950 answers
Step 1
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{x}}}{{{y}}}}\)
we can rewrite it in differential notation and integrate:
xdx=ydy
\(\displaystyle\int{x}{\left.{d}{x}\right.}=\int{y}{\left.{d}{y}\right.}\)
\(\displaystyle{\frac{{{1}}}{{{2}}}}{x}^{{2}}+{c}={\frac{{{1}}}{{{2}}}}{y}^{{2}}\)
\(\displaystyle{y}^{{2}}={x}^{{2}}+{C}\)
\(\displaystyle{y}=\pm\sqrt{{{x}^{{2}}+{c}}}\)
square root always positive and initial value of y is negative
\(\displaystyle{y}=-\sqrt{{{x}^{{2}}+{C}}}\)
Since y(0)=-3
\(\displaystyle{y}{\left({0}\right)}=-\sqrt{{{0}^{{2}}+{C}}}=-{3}\)
C=9
\(\displaystyle{y}=-\sqrt{{{x}^{{2}}+{9}}}\)
Result
\(\displaystyle{y}=-\sqrt{{{x}^{{2}}+{9}}}\)
Have a similar question?
Ask An Expert
0
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-10-29
Find the particular solution that satisfies the differential equation and the initial condition. \(\displaystyle{f}”{\left({x}\right)}={\sin{{x}}},{f}’{\left({0}\right)}={1},{f{{\left({0}\right)}}}={6}\)
asked 2021-03-21
Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={4}{e}^{{-{3}{x}}}\)
asked 2021-10-28
Show that the equation has exactly one real root.
\(\displaystyle{2}{x}+{\cos{{x}}}={0}\)
asked 2021-05-28
Given that \(\displaystyle{f{{\left({x}\right)}}}={\cos{{x}}}\), show that \(\displaystyle\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}={\cos{{x}}}{\left(\frac{{{\text{cosh}{-}}{1}}}{{h}}\right)}+{\sin{{x}}}{\left(\frac{{\text{sinh}}}{{h}}\right)}\)
asked 2021-05-02
Given that \(\displaystyle{t}{\left({x}\right)}={\sin{{x}}}\), show that \(\displaystyle{f{{\left({x}+{h}\right)}}}-\frac{{f{{\left({x}\right)}}}}{{h}}={\sin{{x}}}{\left(\frac{{{\text{cosh}{-}}{1}}}{{h}}\right)}+{\cos{{x}}}{\left(\frac{{\text{sinh}}}{{h}}\right)}\)
asked 2021-10-29
Find the exact value of the trigonometric function at the given real number.
\(\displaystyle{\sin{{\frac{{{7}\pi}}{{{4}}}}}}\)
asked 2021-10-22
Find the exact value of the trigonometric function at the given real number.
\(\displaystyle{\sin{{\frac{{{5}\pi}}{{{4}}}}}}\)

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...