Answer
The solution to the differential equation is:
\[
y^2 (2 \ln y - 1) = 4 e^x (x - 1) + C
\]
where \( C \) is the constant of integration.
Solution
To solve the differential equation:
\[
\frac{y}{x} \frac{dy}{dx} = \frac{e^{x}}{\ln y}
\]
we can use a substitution method. Here's a step-by-step solution:
### Step 1: Rewrite the Equation
Start by rewriting the equation for clarity:
\[
\frac{y}{x} \frac{dy}{dx} = \frac{e^{x}}{\ln y}
\]
### Step 2: Introduce a Substitution
Let’s make the substitution:
\[
u = \ln y \quad \Rightarrow \quad y = e^u
\]
Differentiate both sides with respect to \( x \):
\[
\frac{dy}{dx} = e^u \frac{du}{dx}
\]
### Step 3: Substitute into the Original Equation
Substitute \( y = e^u \) and \( \frac{dy}{dx} = e^u \frac{du}{dx} \) into the original equation:
\[
\frac{e^u}{x} \cdot e^u \frac{du}{dx} = \frac{e^{x}}{u}
\]
Simplify:
\[
\frac{e^{2u}}{x} \frac{du}{dx} = \frac{e^{x}}{u}
\]
Rearrange to separate variables:
\[
u e^{2u} \, du = x e^{x} \, dx
\]
### Step 4: Integrate Both Sides
Integrate both sides:
\[
\int u e^{2u} \, du = \int x e^{x} \, dx
\]
**Left Side Integration:**
Let \( t = 2u \), then \( du = \frac{dt}{2} \):
\[
\int u e^{2u} \, du = \frac{1}{4} e^{2u} (2u - 1) + C_1
\]
**Right Side Integration:**
Use integration by parts where \( a = x \) and \( db = e^x dx \):
\[
\int x e^{x} \, dx = e^x (x - 1) + C_2
\]
### Step 5: Combine the Results
Set the integrals equal to each other:
\[
\frac{1}{4} e^{2u} (2u - 1) = e^x (x - 1) + C
\]
Multiply both sides by 4 to simplify:
\[
e^{2u} (2u - 1) = 4 e^x (x - 1) + C'
\]
### Step 6: Substitute Back for \( u \)
Recall that \( u = \ln y \):
\[
e^{2 \ln y} (2 \ln y - 1) = 4 e^x (x - 1) + C'
\]
Simplify \( e^{2 \ln y} = y^2 \):
\[
y^2 (2 \ln y - 1) = 4 e^x (x - 1) + C
\]
### Final Implicit Solution
The solution to the differential equation is given implicitly by:
\[
y^2 (2 \ln y - 1) = 4 e^x (x - 1) + C
\]
where \( C \) is the constant of integration.
**Note:** Solving for \( y \) explicitly in terms of \( x \) is generally not straightforward for this equation, so the implicit form is the most practical representation of the solution.
**Answer:**
After simplifying, an implicit solution is y²(2 ln y – 1) equals four eˣ(x – 1) plus a constant. Thus,
y²(2 ln y − 1) = 4 e^x(x−1) + C
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
