If $(a+b x) e^{y / x}=x$ then $x^{3} \frac{d^{2} y}{d x^{2}}=$

Question

Professor PC Thomas Classes · Answer

$(a+b x) e^{y / x}=x$\ $\Rightarrow \quad e^{y / x}=\frac{x}{a+b x}$\ Diff. w.r.t. $x$, we get,\ $e^{y / x} \frac{\left[(x) \frac{d y}{d x}-yight]}{x^{2}}=\frac{a+b x-b x}{(a+b x)^{2}}$\ or $\left(x \frac{d y}{d x}-yight) e^{y / x}=\frac{a x^{2}}{(a+b x)^{2}}$\ From (1) using $e^{y / x}=\frac{x}{a+b x}$, we get $\left(x \frac{d y}{d x}-yight)=\frac{a x}{a+b x}$\ Differentiating (3) w.r.t. $x$, we get\ $x \frac{d^{2} y}{d x^{2}}+\frac{d y}{d x}-\frac{d y}{d x}=\frac{(a+b x) a-a x b}{(a+b x)^{2}}$\ or $x \frac{d^{2} y}{d x^{2}}=\frac{a^{2}}{(a+b x)^{2}}$\ Multiplying by $x^{2}$ and using (3) in R.H.S.\ $\Rightarrow \quad x^{3} \frac{d^{2} y}{d x^{2}}=\left(x \frac{d y}{d x}-yight)^{2}$\