Solve the given differential equation. \frac{dy}{dx}=\frac{y}{x}

Cem Hayes 2021-10-04 Answered
Solve the given differential equation.
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{y}}}{{{x}}}}\)

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

Willie
Answered 2021-10-05 Author has 17235 answers
Step 1
We have to solve the differential equation:
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{y}}}{{{x}}}}\)
Solving this equation by variable separable method(shifting same variable with same derivatives),
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{{y}}}{{{x}}}}\)
\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{y}}}}={\frac{{{\left.{d}{x}\right.}}}{{{x}}}}\)....(1)
Step 2
Integrating equation (1) both sides, we get
\(\displaystyle\int{\frac{{{\left.{d}{y}\right.}}}{{{y}}}}=\int{\frac{{{\left.{d}{x}\right.}}}{{{x}}}}\)
\(\displaystyle{\ln{{\left({y}\right)}}}={\ln{{\left({x}\right)}}}+{\ln{{\left({c}\right)}}}\)
(Using formula \(\displaystyle\int{\frac{{{1}}}{{{x}}}}{\left.{d}{x}\right.}={\ln{{\left({x}\right)}}}+{c}\))
(Here we have used ln(c) as an arbitrary constant due to all terms in logarithmic)
Now, using the property of logarithmic \(\displaystyle{\log{{\left({a}\right)}}}+{\log{{\left({b}\right)}}}={\log{{\left({a}{b}\right)}}}\ {\quad\text{and}\quad}\ {\log{{\left({a}\right)}}}={\log{{\left({b}\right)}}}\Rightarrow{a}={b}\), we get
\(\displaystyle{\ln{{\left({y}\right)}}}={\ln{{\left({x}\right)}}}+{\ln{{\left({c}\right)}}}\)
\(\displaystyle{\ln{{\left({y}\right)}}}={\ln{{\left({x}{c}\right)}}}\)
y=xc
\(\displaystyle{c}={\frac{{{y}}}{{{x}}}}\)
Hence, solution of differential given equation is \(\displaystyle{c}={\frac{{{y}}}{{{x}}}}\).
Have a similar question?
Ask An Expert
25
 

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
...