Question

Find the solution of the differential equation \[9y\left( {\dfrac{{dy}}{{dx}}} \right) + 4x = 0\]
A. \[\left( {\dfrac{{{y^2}}}{9}} \right) + \left( {\dfrac{{{x^2}}}{4}} \right) = c\]
B. \[\left( {\dfrac{{{y^2}}}{4}} \right) + \left( {\dfrac{{{x^2}}}{9}} \right) = c\]
C. \[\left( {\dfrac{{{y^2}}}{9}} \right) - \left( {\dfrac{{{x^2}}}{4}} \right) = c\]
D. \[\left( {{y^2}} \right) - \left( {\dfrac{{{x^2}}}{9}} \right) = c\]

Answer 1

Hint: The solution of a differential equation is the relationship between the variables of a differential equation that fulfils the specified differential equation. In this question, the solutions of the differential equation can be obtained by integrating the differential equation by separating \[y\] and \[dy\] and \[x\] and \[dx\] terms.

Formula used: The following formula of integration can be used to solve this example.
\[\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \]

Complete step-by-step solution: We know that the given differential equation is \[9y\left( {\dfrac{{dy}}{{dx}}} \right) + 4x = 0\]
Let us simplify the above differential equation.
Thus, we get
\[9y\left( {\frac{{dy}}{{dx}}} \right) = - 4x\]
\[ \Rightarrow 9ydy = - 4xdx\]
By integrating on both sides, we get
\[\int {9ydy} = \int { - 4xdx} \]
\[ \Rightarrow 9\int {ydy} = - 4\int {xdx} \]
But we know that \[\int {{x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \]
So, we get
\[
   \Rightarrow 9\left( {\dfrac{{{y^2}}}{2}} \right) = - 4\left( {\dfrac{{{x^2}}}{2}} \right) + c' \\
   \Rightarrow \left( {\dfrac{{9{y^2}}}{2}} \right) + \left( {\dfrac{{4{x^2}}}{2}} \right) = c' \\
   \Rightarrow \left( {\dfrac{{9{y^2} + 4{x^2}}}{2}} \right) = c' \\
   \Rightarrow \left( {9{y^2} + 4{x^2}} \right) = 2c'
 \]
Here, \[c'\] is a constant of integration.
Now, take LCM (Least Common Multiple) of \[9\] and \[4\] is \[36\]
Thus, divide by \[36\] on both sides.
So, we get
\[
  \left( {\dfrac{{9{y^2}}}{{36}} + \dfrac{{4{x^2}}}{{36}}} \right) = \dfrac{{2c'}}{{36}} \\
   \Rightarrow \left( {\dfrac{{{y^2}}}{4} + \dfrac{{{x^2}}}{9}} \right) = \dfrac{{c'}}{{18}} \\
 \]
Put \[\dfrac{{c'}}{{18}} = c\] in the above equation.
Thus, we get
\[\left( {\dfrac{{{y^2}}}{4} + \dfrac{{{x^2}}}{9}} \right) = c\]
Hence, the solution of the differential equation \[9y\left( {\dfrac{{dy}}{{dx}}} \right) + 4x = 0\] is \[\left( {\dfrac{{{y^2}}}{4} + \dfrac{{{x^2}}}{9}} \right) = c\]

Therefore, the option (B) is correct.

Additional Information: The differential equation is defined as an equation that consists of one or more functions and their derivatives. There are several methods to solve differential equations such as variable separable method, inspection method, homogeneous method, etc.

Note: There are two types of solutions to differential equations such as general solution and particular solution. A general solution of the nth order differential equation is the solution that contains vital arbitrary constants. Whereas, a particular solution of a differential equation is one generated from the general solution by assigning specific values to an arbitrary solution.