Problem 28 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The general solution of the differential equation \(\left(y^{2}-x^{3}\right)\) \(\mathrm{d} x-x y d y=0(x \neq 0)\) is: (a) \(y^{2}-2 x^{2}+c x^{3}=0\) (b) \(y^{2}+2 x^{3}+c x^{2}=0\) (c) \(y^{2}+2 x^{2}+c x^{3}=0\) (d) \(y^{2}-2 x^{3}+c x^{2}=0\) (where \(c\) is a constant of integration)

Step-by-Step Solution

Answer

The correct answer is (c) \( y^2 + 2x^2 + cx^3 = 0 \).

Step 1: Rewrite the Differential Equation

The given differential equation is \( (y^2 - x^3)\, dx - x y \, dy = 0 \). We need to formulate it in a standard separable or solvable form. Let us write the equation as: \( (y^2 - x^3)\, dx = x y \, dy \).

Step 2: Separate Variables

We aim to separate the variables. Rearrange the equation to isolate terms involving similar variables. Divide both sides by \(x(y^2 - x^3)\) to obtain: \( \frac{dx}{x} = \frac{y \, dy}{y^2 - x^3} \).

Step 3: Introduce Integrating Factor

Since separating directly might still be complicated, multiply and divide by \(x^3\) to help with integration: The equation becomes: \(-\frac{dy}{y} + \frac{dy}{x^3} = -\frac{dx}{x^3} + \frac{dx}{y^2} \). Rearranging further with integrating factors will simplify this integral.

Step 4: Integrate Both Sides

The integration solution combines terms: \( \int x^2 \,dx + \int \frac{1}{y^2} \, dy = \int 2 x \, dx \). Integrate each side: Solving it gives \( y^2 = x^2 + c x^3 \).

Step 5: Simplify and Compare Solutions

Simplify the equation obtained: Split into: \( y^2 + 2x^3 + cx^2 = 0 \). Compare with the answer options: Match with answer (c).

Key Concepts

Integration Techniques

Understanding integration techniques is key when solving differential equations, like the one given in the problem. These techniques involve finding the antiderivatives or integrals of functions. One common approach used in solving equations like this one is to set up the problem so that you can identify and apply a fitting technique that will simplify and solve the equation effectively.

For instance, in the original problem, we can manage the complexity by using substitution and manipulation of terms. These methods help transform difficult integrals into simpler forms. We often make use of power rule integration, where we acknowledge that the integral of a power of x, say \( x^n \), is \( \frac{x^{n+1}}{n+1} \) plus a constant, as long as \( n eq -1 \). Such techniques make it possible to integrate each term separately:

• You solve for \( \int x^2 \, dx \)
• You solve for \( \int \frac{1}{y^2} \, dy \)

This segmentation into simpler integration parts is what enables us to handle more complicated differential equations.

Separating Variables

The concept of separating variables is crucial in dealing with differential equations. This technique works well when the equation can be written in a form that allows variables to be isolated on either side of the equation. The idea is to rearrange the terms of the differential equation so that all terms involving one variable and its differentials are on one side, and all terms involving the other variable and its differentials are on the opposite side.
For the equation \( (y^2 - x^3)\, dx = x y \, dy \), separating variables involves dividing or multiplying both sides by functions to isolate the variables x and y:

• This led us to \( \frac{dx}{x} = \frac{y \, dy}{y^2 - x^3} \).
• This isolated the terms, making it so we can individually integrate both sides with respect to their variables.

Separation of variables thus simplifies the equation and allows us to solve integrals separately, one involving \( x \) and the other involving \( y \). This technique is a powerful tool in solving many first-order differential equations.

Integrating Factors

Integrating factors are an invaluable tool when you face linear differential equations that are hard to solve. The purpose of an integrating factor is to turn a non-exact equation into an exact one. An exact equation is one where the differential can be expressed as the differential of a function of only one variable, making it easier to solve.

However, in the context of the original differential equation, the use of integrating factors was implied by simplifying the rearranged terms after separation. When the equation \(-\frac{dy}{y} + \frac{dy}{x^3} = -\frac{dx}{x^3} + \frac{dx}{y^2} \) happens, multiplying by an integrating factor allows it to become more straightforward to approach:

• It helps balance the equation in simplifying integration.
• The result is that all parts are manageable in terms of finding antiderivatives.

This method acts as an assisting factor in reaching a final, integrably smooth form that ultimately leads to a solution.