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 \(\displaystyle{c}={\frac{{{y}}}{{{x}}}}\).

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 \(\displaystyle{c}={\frac{{{y}}}{{{x}}}}\).