Question
(x) \( \frac{y}{x} \frac{d y}{d x}=\frac{e^{x}}{\ln y} \)
Ask by Wilson Zimmerman. in Egypt
Dec 24,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution to the differential equation is:
\[
\frac{y^2}{2} \ln y - \frac{y^2}{4} = e^{x}(x - 1) + C
\]
where \( C \) is the constant of integration.
Solution
To solve the differential equation:
\[
\frac{y}{x} \frac{dy}{dx} = \frac{e^{x}}{\ln y}
\]
Let's follow the steps to find an implicit solution for \( y \) in terms of \( x \).
### Step 1: Rewrite the Equation
First, isolate \( \frac{dy}{dx} \):
\[
\frac{dy}{dx} = \frac{e^{x}}{\ln y} \cdot \frac{x}{y}
\]
This simplifies to:
\[
\frac{dy}{dx} = \frac{e^{x} \cdot x}{y \cdot \ln y}
\]
### Step 2: Separate Variables
Rearrange the equation to separate the variables \( y \) and \( x \):
\[
y \cdot \ln y \cdot dy = e^{x} \cdot x \cdot dx
\]
### Step 3: Integrate Both Sides
Now, integrate both sides of the equation.
**Left Side:**
\[
\int y \cdot \ln y \, dy
\]
Use integration by parts where:
- Let \( u = \ln y \) ⇒ \( du = \frac{1}{y} dy \)
- Let \( dv = y \, dy \) ⇒ \( v = \frac{y^2}{2} \)
Applying integration by parts:
\[
\int y \cdot \ln y \, dy = \frac{y^2}{2} \ln y - \int \frac{y^2}{2} \cdot \frac{1}{y} \, dy = \frac{y^2}{2} \ln y - \frac{y^2}{4} + C_1
\]
**Right Side:**
\[
\int e^{x} \cdot x \, dx
\]
Again, use integration by parts where:
- Let \( u = x \) ⇒ \( du = dx \)
- Let \( dv = e^{x} \, dx \) ⇒ \( v = e^{x} \)
Applying integration by parts:
\[
\int e^{x} \cdot x \, dx = e^{x} \cdot x - \int e^{x} \, dx = e^{x} \cdot x - e^{x} + C_2
\]
### Step 4: Combine the Results
Equate the integrals from both sides:
\[
\frac{y^2}{2} \ln y - \frac{y^2}{4} = e^{x} \cdot x - e^{x} + C
\]
where \( C = C_2 - C_1 \) is the constant of integration.
### Step 5: Simplify (Optional)
You can multiply the entire equation by 4 to eliminate denominators:
\[
2y^2 \ln y - y^2 = 4e^{x} (x - 1) + C'
\]
where \( C' = 4C \) is a new constant of integration.
### Final Implicit Solution
The implicit solution to the differential equation is:
\[
\frac{y^2}{2} \ln y - \frac{y^2}{4} = e^{x}(x - 1) + C
\]
where \( C \) is the constant of integration.
**Note:** Solving explicitly for \( y \) in terms of \( x \) is generally difficult for this equation, so the solution is typically left in this implicit form.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve the given differential equation \( \frac{y}{x} \frac{d y}{d x}=\frac{e^{x}}{\ln y} \), it's important to understand the approach to separable equations. Rearranging terms could lead us to separate variables, allowing integration on both sides. Also, remember that when dealing with logarithmic functions, it’s critical to check the domain of the solution since \(\ln y\) is only defined for \(y > 0\). This can often lead to additional constraints on \(y\) based on the context of the problem you're solving!
