Obtain the differential equation by eliminating arbitrary constan… - Vidyadip

Question

Obtain the differential equation by eliminating arbitrary constants from the following equations : $y^2=(x+c)^3$

✓

Answer

$
y^2=(x+c)^3
$
Differentiating w.r.t. $x$, we get
$
\begin{aligned}
& 2 y \frac{d y}{d x}=3(x+c)^2 \cdot(1)=3(x+c)^2 \\
& \therefore(x+c)^2=\frac{2 y}{3} \cdot \frac{d y}{d x} \\
& \therefore(x+c)^6=\left(\frac{2 y}{3} \cdot \frac{d y}{d x}\right)^3 \\
& \therefore\left(y^2\right)^2=\frac{8 y^3}{27} \cdot\left(\frac{d y}{d x}\right)^3 \quad \ldots \ldots . .[ By (1)]
\end{aligned}
$
$
\begin{aligned}
& \therefore 27 y^4=8 y^3\left(\frac{d y}{d x}\right)^3 \\
& \therefore 27 y=8\left(\frac{d y}{d x}\right)^3 \\
& \therefore\left(\frac{d y}{d x}\right)^3=\frac{27 y}{8} \\
& \therefore \frac{d y}{d x}=\frac{3}{2}(\sqrt[3]{y})
\end{aligned}
$
This is the required D.E.

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