Question

What is a solution to the differential equation $\dfrac{dy}{dx}=xy$?

Answer 1

To obtain the solution of the differential equation given use separable of variable formula. Firstly we will separate the variables in the equation and get them together with their respective derivatives. Then we will integrate the equation on both sides with respect to $x$. Finally we will solve our integration sign and get the desired answer.

Complete step-by-step solution:
To differential equation given is:
$\dfrac{dy}{dx}=xy$…….$\left( 1 \right)$
We will use separable of variable formula in above equation by taking all the $x$ term with its derivative on one side and all the $y$ terms with its derivative on other side as:
$\dfrac{1}{y}\times dy=x\times dx$
Now we will integrate the above equation by using the formula given below:
$\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}+C$
$\int{\dfrac{1}{y}dy=\ln y}$
So we get,
$\begin{aligned} & \Rightarrow \int{\dfrac{1}{y}\times dy}=\int{x\times dx} \\\ & \Rightarrow \ln y=\dfrac{{{x}^{2}}}{2}+C \\\ \end{aligned}$
Now, taking logarithm from left side to right side we get,
$\begin{aligned} & \Rightarrow y={{e}^{\left( \dfrac{{{x}^{2}}}{2}+C \right)}} \\\ & \Rightarrow y={{e}^{\left( \dfrac{{{x}^{2}}}{2} \right)}}\times {{e}^{C}} \\\ & \therefore y=A{{e}^{\dfrac{{{x}^{2}}}{2}}} \\\ \end{aligned}$
Where $A={{e}^{c}}$
Hence solution of differential equation $\dfrac{dy}{dx}=xy$ is $A{{e}^{\dfrac{{{x}^{2}}}{2}}}$.

Note: A differential equation is the one which relates one or more function and derivative of one function i.e. dependent variable with respect to the other variable i.e. independent variable. The function used represents physical quantities whereas the derivatives represent the rate of change. The separation of variable is used when the given differential equation is of the form $\dfrac{dy}{dx}=f\left( x \right)h\left( y \right)$such that the $x,y$ variable can be separated easily. Ti is widely used in many fields of science like in physics, engineering and even in biology. Logarithm function is an inverse function of exponential function; it is defined as $y={{\log }_{b}}x$ where $b$ is the base. The logarithm function with base 10 is known as natural logarithm.