Question

What is the general solution of the differential equation \[\dfrac{{dy}}{{dx}} = \dfrac{{{x^2}}}{{{y^2}}}\]?
A. \[{x^3} - {y^3} = c\]
B. \[{x^3} + {y^3} = c\]
C. \[{x^2} + {y^2} = c\]
D. \[{x^2} - {y^2} = c\]

Answer 1

Hint: Here, the first order differential equation is given. First, simplify the given equation by rearranging the terms. Then, integrate both sides of the equation with respect to the corresponding variables. In the end, solve the integrals by using the standard integration formula and get the general solution of the differential equation.

Formula Used: \[\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\]

Complete step by step solution: The given differential equation is \[\dfrac{{dy}}{{dx}} = \dfrac{{{x^2}}}{{{y^2}}}\].

Simplify the given equation by cross multiplying the denominators.
\[{y^2}dy = {x^2}dx\]
Now integrate both sides with respect to the corresponding variables.
\[\int {{y^2}dy} = \int {{x^2}dx} \]
Apply the integration formula \[\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\] on both sides.
We get,
\[\dfrac{{{y^3}}}{3} = \dfrac{{{x^3}}}{3} + \dfrac{{{c^3}}}{3}\]
Simplify the equation.
Multiply both sides by 3.
\[ Rightarrow {y^3} = {x^3} + {c_1}^3\]
\[ Rightarrow {y^3} = {x^3} + c\]
\[ \Rightarrow {y^3} - {x^3} = c\] or \[{x^3} - {y^3} = c\]
Therefore, the general solution of the differential equation \[\dfrac{{dy}}{{dx}} = \dfrac{{{x^2}}}{{{y^2}}}\] is \[{x^3} - {y^3} = c\].

Option ‘A’ is correct

Note: Students often do mistake to integrating \[\int {{x^n}} dx\]. They apply the formula \[\int {{x^n}} dx = {x^{n + 1}} + c\], which is incorrect formula. They forget to divide the term \[{x^{n + 1}}\] by \[n + 1\] The correct formula is \[\int {{x^n}} dx = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\].