The general solution of the differential equation dy/dx=y/x is given by y=kx where k is any constant. In this post, we will learn how to find the solution of dy/dx=y/x.

General Solution of dy/dx=y/x
Question: Find the general solution of $\dfrac{dy}{dx}=\dfrac{y}{x}$.
Answer:
$\dfrac{dy}{dx}=\dfrac{y}{x}$
We can rewrite it as
$\dfrac{dy}{y}=\dfrac{dx}{x}$
Integrating both sides, we get that
$\int \dfrac{dy}{y}=\int \dfrac{dx}{x} +C$ where C is a constant of integration.
⇒ ln|y| = ln|x| + ln|k| where ln is the natural logarithm, that is, ln=loge and we replace C with another constant ln|k|
⇒ ln|y| = ln|kx| by the logarithm rules
⇒ y=kx
So the general solution of dy/dx=y/x is equal to y=kx where k is any constant.
Particular Solution of dy/dx=y/x
Question: Find the particular solution of $\dfrac{dy}{dx}=\dfrac{y}{x}, y(1)=1$.
Answer:
We know that y=kx is the general solution of dy/dx=y/x.
Given y(1)=1, that is, x=1, y=1.
So from y=kx we have
1=k ⋅ 1
⇒ k=1.
So y=x is the particular solution of dy/dx=y/x when y(1)=1.
FAQs
Q: What is the general solution of dy/dx=y/x?
Answer: The general solution of dy/dx=y/x is y=cx where c is an arbitrary constant.