Solve the initial value problem for y as a function of x.
-$$\left( x ^ { 2 } + 81 \right) ^ { 2 } \frac { d y } { d x } = \sqrt { x ^ { 2 } + 81 } , y ( 0 ) = 9$$
A) $y = \frac { x } { 9 } + 9$
B) $y = \frac { x } { 81 } + 9$
C) $y = \frac { x } { 81 \sqrt { x ^ { 2 } + 81 } } + 9$
D) $y = \frac { x } { 81 \sqrt { x ^ { 2 } + 81 } }$

Question

Quizplus · Accepted Answer

We begin by separating variables and integrating both sides:
\begin{align*}
\left(x^{2}+81\right)^{2} \frac{dy}{dx} &=\sqrt{x^{2}+81} \
\frac{1}{\left(x^{2}+81\right)^{2}} d y &=\frac{1}{\sqrt{x^{2}+81}} d x \
\int \frac{1}{\left(x^{2}+81\right)^{2}} d y &=\int \frac{1}{\sqrt{x^{2}+81}} d x \
-\frac{1}{81} \frac{1}{\sqrt{x^{2}+81}}+C_{1} &=\ln \left|\frac{x}{9}+\sqrt{x^{2}+81}\right|+C_{2}
\end{align*}
Using the initial condition $y(0)=9$, we have
$$C_{1}=\frac{1}{81 \cdot 9}=\frac{1}{729}$$
$$\ln \left|\frac{0}{9}+\sqrt{0^{2}+81}\right|+C_{2}=\ln 9+C_{2}$$
So we have
$$-\frac{1}{81} \frac{1}{\sqrt{x^{2}+81}}=\ln \left|\frac{x}{9}+\sqrt{x^{2}+81}\right|-\frac{1}{729}+\ln 9$$
$$y(x)=\frac{1}{81 \sqrt{x^{2}+81}}\left(729 \ln 9-9 \ln \left|\frac{x}{9}+\sqrt{x^{2}+81}\right|\right)+9$$
$$y(x)=\frac{1}{81 \sqrt{x^{2}+81}}\left(729 \ln \left|\frac{81}{x+\sqrt{x^{2}+81}}\right|\right)+9$$
$$y(x)=\frac{x}{81 \sqrt{x^{2}+81}}+9-\frac{1}{81 \sqrt{x^{2}+81}}\left(729 \ln \left|\frac{x+\sqrt{x^{2}+81}}{9}\right|\right)$$
We can simplify the expression $\frac{729 \ln \left|\frac{x+\sqrt{x^{2}+81}}{9}\right|}{81 \sqrt{x^{2}+81}}$ as follows:
$$\frac{729 \ln \left|\frac{x+\sqrt{x^{2}+81}}{9}\right|}{81 \sqrt{x^{2}+81}}=\frac{9 \cdot 81 \ln \left|\frac{x+\sqrt{x^{2}+81}}{9}\right|}{9 \cdot \sqrt{x^{2}+81}}=9\ln \left|\frac{x+\sqrt{x^2+81}}{9}\right|\cdot \frac{\sqrt{x^2+81}}{9(x^2+81)}$$
So the final answer is
$$y(x)=\frac{x}{81 \sqrt{x^{2}+81}}+9-\frac{\sqrt{x^{2}+81}}{9 x^{2}+729} \cdot 9 \ln \left|\frac{x+\sqrt{x^{2}+81}}{9}\right|$$
which matches choice C.

Solve the Initial Value Problem for Y as a Function (x2+81)2dydx=x2+81,y(0)=9\left( x ^ { 2 } + 81 \right) ^ { 2 } \frac { d y } { d x } = \sqrt { x ^ { 2 } + 81 } , y ( 0 ) = 9(x2+81)2dxdy​=x2+81​,y(0)=9

Solve the Initial Value Problem for Y as a Function $\left( x ^ { 2 } + 81 \right) ^ { 2 } \frac { d y } { d x } = \sqrt { x ^ { 2 } + 81 } , y ( 0 ) = 9$