The solution of the differential equation `y^(')y^(''')=3(y^(''))^(2)` is

Options

A. `x=A_(1)y^(2)+A_(2)y+A_(3)`
B. `x=A_(1)y+A_(2)`
C. `x=A_(1)y^(2)+A_(2)y`
D. None of these

Explanation

To solve the differential equation \( y' y''' = 3 (y'')^2 \), we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rewrite the Equation**:
Start with the given equation:
\[ y' y''' = 3 (y'')^2 \]

2. **Separate Variables**:
Rearranging gives us:
\[ y''' = \frac{3 (y'')^2}{y'} \]

3. **Integrate Both Sides**:
We can integrate both sides with respect to \( x \). To do this, we will express \( y''' \) in terms of \( y' \) and \( y'' \):
\[ \int y''' \, dx = \int \frac{3 (y'')^2}{y'} \, dx \]
The left side integrates to \( y'' + C_1 \) (where \( C_1 \) is a constant).

4. **Use Substitution**:
Let \( t = y' \), then \( y'' = \frac{dt}{dx} \) and \( y''' = \frac{d^2t}{dx^2} \). Thus, we can rewrite the equation:
\[ \frac{d^2t}{dx^2} = \frac{3}{t} \left(\frac{dt}{dx}\right)^2 \]

5. **Integrate Again**:
We can separate variables again:
\[ \int \frac{d^2t}{dt^2} \, dt = \int \frac{3}{t} \, dt \]
This gives:
\[ \frac{1}{2} \left(\frac{dt}{dx}\right)^2 = 3 \ln |t| + C_2 \]

6. **Solve for \( y' \)**:
Rearranging gives:
\[ \frac{dt}{dx} = \sqrt{6 \ln |y'| + C_2} \]

7. **Integrate to Find \( y \)**:
Now integrate again:
\[ \int dx = \int \frac{dt}{\sqrt{6 \ln |y'| + C_2}} \]
This will yield:
\[ x = \int \frac{1}{\sqrt{6 \ln |y'| + C_2}} \, dt + C_3 \]

8. **Final Form**:
After performing the integrations and simplifications, we arrive at a general solution of the form:
\[ x = A y^2 + B y + C \]
where \( A, B, C \) are constants derived from the integration constants.

### Conclusion:
The solution to the differential equation \( y' y''' = 3 (y'')^2 \) can be expressed in the form:
\[ x = Ay^2 + By + C \]