"Solve the given differential equation. dydx=yx" solved and explained

Cem Hayes

Answered question

2021-10-04

Solve the given differential equation.

\frac{d y}{d x} = \frac{y}{x}

Willie

Skilled2021-10-05Added 95 answers

Step 1
We have to solve the differential equation:

\frac{d y}{d x} = \frac{y}{x}

Solving this equation by variable separable method(shifting same variable with same derivatives),

\frac{d y}{d x} = \frac{y}{x}

\frac{d y}{y} = \frac{d x}{x}

....(1)
Step 2
Integrating equation (1) both sides, we get

\int \frac{d y}{y} = \int \frac{d x}{x}

\ln (y) = \ln (x) + \ln (c)

(Using formula

\int \frac{1}{x} d x = \ln (x) + c

)
(Here we have used ln(c) as an arbitrary constant due to all terms in logarithmic)
Now, using the property of logarithmic

\log (a) + \log (b) = \log (a b) and \log (a) = \log (b) \Rightarrow a = b

, we get

\ln (y) = \ln (x) + \ln (c)

\ln (y) = \ln (x c)

y=xc

c = \frac{y}{x}

Hence, solution of differential given equation is

c = \frac{y}{x}

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