Question

What is a solution to the differential equation \[\dfrac{{dy}}{{dx}} = x - y\] ?

Answer 1

Hint: Here, we need to differentiate the given equation, \[\dfrac{{dy}}{{dx}} = x - y\]. A differential equation includes derivatives, which can be either partial or regular. The differential equation describes a relationship between a quantity that is continuously varying with respect to a change in another quantity, and the derivative represents a rate of change. There are several differential equation formulas for determining derivative solutions.

Complete step by step solution:
In the given problem,
The differential equation is \[\dfrac{{dy}}{{dx}} = x - y\]
We can’t separate this equation, so we have to set up an integrating the factor, we get
  \[\dfrac{{dy}}{{dx}} + y = x\]
Here, the integrating factor is \[{e^{\int {dx} }} = {e^x}\] , we get
 \[{e^x}\dfrac{{dy}}{{dx}} + {e^x}y = x{e^x}\]
So, we take out commonly, \[{e^x}y\] from above, we get
 \[\dfrac{d}{{dx}}({e^x}y) = x{e^x}\]
So, the integration factor, \[{e^x}y = \int {x{e^x}dx} \to (1)\]
 We use the integration by part formula, \[\int {u{v^1} = uv - \int {{u^1}v} } \]
 \[
  u = x,{u^1} = 1 \\
  v' = {e^x},v = {e^x} \;
\]
By substituting the values into the formula, we get
 \[\int {u{v^1} = uv - \int {{u^1}v} } \Rightarrow \int {x{e^x} = x{e^x} - \int {{e^x}dx} } \]
To simplify the integration, we can get
 \[\int {x{e^x}} = x{e^x} - {e^x} + C\]
By substituting the above values in equation \[(1)\] , we get
 \[{e^x}y = x{e^x} - {e^x} + C\]
By dividing on both sides by \[{e^x}\] , we get
 \[y = x - 1 + \dfrac{C}{{{e^x}}}\]
Thus, \[y = x - 1 + \dfrac{C}{{{e^x}}}\] is the solution to the differential equation, \[\dfrac{{dy}}{{dx}} = x - y\] .
So, the correct answer is \[y = x - 1 + \dfrac{C}{{{e^x}}}\] ”.

Note: We note that the differential equation is one that includes one or more terms as well as the derivatives of one variable (the dependent variable) with respect to another variable (i.e., independent variable) \[\dfrac{{dy}}{{dx}} = f(x)\]. Where, the variable, \[x\] is independent and the variable \[y\] is dependent.Thus, the differentiation is a process of finding a function that outputs the rate of change of one variable with respect to another variable. We use the integrating factor formula, \[\int {u{v^1} = uv - \int {{u^1}v} } \] is used as the derivative of a function. The function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument.