Question

Eliminate constants from $y=A(x+B)^2$

• A. $2yy''=(y')^2$
• B. $yy''=2y'$
• C. $2yy''=y'+y$
• D. $2yy''=y'-y$

Hint:

Divide derivative equations to eliminate constants quickly.

Answer 1

The Correct Option is A

Step 1: Know the goal.
We have $y=A(x+B)^2$ with two unknown constants $A$ and $B$. To form a differential equation we must remove both constants by differentiating.

Step 2: Differentiate once.
\[ y'=2A(x+B). \] This still has $A$ and $B$ in it.

Step 3: Differentiate again.
\[ y''=2A. \] Now $y''$ contains only $A$, which is helpful.

Step 4: Express $(x+B)$ neatly.
From $y=A(x+B)^2$ and $y'=2A(x+B)$, divide to cancel $A$: \[ \frac{2y}{y'}=\frac{2A(x+B)^2}{2A(x+B)}=(x+B). \]
Step 5: Substitute into $y'$.
Put $(x+B)=\dfrac{2y}{y'}$ into $y'=2A(x+B)$. Since $2A=y''$, we get \[ y'=y''\cdot\frac{2y}{y'}. \]
Step 6: Clean up.
Multiply both sides by $y'$: \[ (y')^2=2y\,y''. \] So the required differential equation is $2yy''=(y')^2$.
\[ \boxed{2yy''=(y')^2} \]