What is the solution of the differential equation \(\ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) - {\rm{a}} = 0?\)
y = xe a+ c
Solving the Differential Equation: Step-by-Step Guide
We are asked to find the solution of the differential equation given by: \[ \ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) - {\rm{a}} = 0 \] This is a first-order differential equation. To solve it, we need to isolate the derivative term \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\) and then integrate.
Isolating the Derivative Term
The first step is to move the constant 'a' to the right side of the equation:
\[ \ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) = {\rm{a}} \]The natural logarithm function, \(\ln(x)\), is the inverse of the exponential function, \(e^x\). To eliminate the natural logarithm on the left side, we can exponentiate both sides of the equation with base 'e':
\[ e^{\ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)} = e^{\rm{a}} \]Using the property \(e^{\ln(x)} = x\), the left side simplifies to \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\):
\[ \frac{{{\rm{dy}}}}{{{\rm{dx}}}} = e^{\rm{a}} \]Here, \(e^{\rm{a}}\) is a constant because 'a' is a constant.
Separating Variables and Integrating
The equation is now in the form \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = \text{constant}\), which is a very simple separable differential equation. We can separate the variables 'y' and 'x' by multiplying both sides by \({\rm{dx}}\):
\[ {\rm{dy}} = e^{\rm{a}} {\rm{dx}} \]Now, we integrate both sides of the equation:
\[ \int {\rm{dy}} = \int e^{\rm{a}} {\rm{dx}} \]Since \(e^{\rm{a}}\) is a constant, we can take it out of the integral on the right side:
\[ \int {\rm{dy}} = e^{\rm{a}} \int {\rm{dx}} \]Performing the integration:
The integral of \({\rm{dy}}\) is \({\rm{y}}\) plus an integration constant, say \({\rm{C}}_1\).
\[ \int {\rm{dy}} = {\rm{y}} + {\rm{C}}_1 \]The integral of \({\rm{dx}}\) is \({\rm{x}}\) plus an integration constant, say \({\rm{C}}_2\).
\[ \int {\rm{dx}} = {\rm{x}} + {\rm{C}}_2 \]Substituting these back into the integrated equation:
\[ {\rm{y}} + {\rm{C}}_1 = e^{\rm{a}} ({\rm{x}} + {\rm{C}}_2) \] \[ {\rm{y}} + {\rm{C}}_1 = e^{\rm{a}} {\rm{x}} + e^{\rm{a}} {\rm{C}}_2 \]Rearranging the terms to solve for 'y':
\[ {\rm{y}} = e^{\rm{a}} {\rm{x}} + e^{\rm{a}} {\rm{C}}_2 - {\rm{C}}_1 \]Since \(e^{\rm{a}}\) is a constant and \({\rm{C}}_1\) and \({\rm{C}}_2\) are constants, the combination \(e^{\rm{a}} {\rm{C}}_2 - {\rm{C}}_1\) is also a constant. Let's call this new constant 'c'.
\[ {\rm{y}} = e^{\rm{a}} {\rm{x}} + {\rm{c}} \]This can also be written as:
\[ {\rm{y}} = {\rm{xe}}^{\rm{a}} + {\rm{c}} \]This is the general solution to the given differential equation.
Comparing with Options
Let's compare our derived solution \({\rm{y}} = {\rm{xe}}^{\rm{a}} + {\rm{c}}\) with the given options:
- Option 1: y = xea + c
- Option 2: x = yea + c
- Option 3: y = ln x + c
- Option 4: x = ln y + c
Our solution matches Option 1.
Revision Table: Differential Equation Solution
| Step | Action | Result |
|---|---|---|
| 1 | Original Equation | \(\ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) - {\rm{a}} = 0\) |
| 2 | Isolate ln term | \(\ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) = {\rm{a}}\) |
| 3 | Exponentiate both sides | \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = e^{\rm{a}}\) |
| 4 | Separate variables | \({\rm{dy}} = e^{\rm{a}} {\rm{dx}}\) |
| 5 | Integrate both sides | \(\int {\rm{dy}} = \int e^{\rm{a}} {\rm{dx}}\) |
| 6 | Perform Integration | \({\rm{y}} = e^{\rm{a}} {\rm{x}} + {\rm{c}}\) |
| 7 | Final Solution | \({\rm{y}} = {\rm{xe}}^{\rm{a}} + {\rm{c}}\) |
Additional Information: Solving Differential Equations
A differential equation is an equation that relates a function with its derivatives. Solving a differential equation means finding the function that satisfies the equation.
- Order of a Differential Equation: The order is the highest order of derivative appearing in the equation. The given equation \(\ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) - {\rm{a}} = 0\) is a first-order differential equation because the highest derivative is the first derivative, \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}}\).
- Separable Differential Equations: A first-order differential equation of the form \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = f(x)g(y)\) is called separable. If \(g(y) \neq 0\), we can separate variables by writing \(\frac{{{\rm{dy}}}}{g(y)} = f(x){\rm{dx}}\) and then integrate both sides: \(\int \frac{{{\rm{dy}}}}{g(y)} = \int f(x){\rm{dx}}\). In our case, \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} = e^{\rm{a}}\), which can be seen as \(f(x) = e^{\rm{a}}\) and \(g(y) = 1\). This makes it separable.
- General Solution: The solution obtained contains an arbitrary constant (like 'c' in our case). This solution represents a family of functions and is called the general solution. The constant arises from the integration process.
Understanding how to isolate the derivative and identify the type of differential equation (like separable) is crucial for finding its solution. Exponentiating is a common technique used to solve equations involving logarithms, just as taking a logarithm is used for exponential equations.
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