Question

The solution of the equation $y^{2}xdx+ydx-xdy=0$ is (1) $x^2+\frac{x}{y}+c=0$ (2) $\frac{x^2}{2}-\frac{x}{y}+c=0$ (3) $\frac{x^2}{2}+\frac{x}{y}+c=0$ (4) $\frac{x^2}{2}-xy+c=0$

Short answer

Option 3: $\frac{x^2}{2}+\frac{x}{y}+c=0$.

Step-by-step solution

Rewrite the Equation in Standard Form

Given the differential equation: $y^{2}xdx+ydx-xdy=0$. Combine the terms with $dx$ and $dy$ to form: $(y^{2}x+y)dx-xdy=0$.

Rearrange the Equation

Reorder the equation for convenience: $(y^{2}x+y)dx=xdy$.

Separate the Variables

Separate the variables to isolate $x$ on one side and $y$ on the other side: $\frac{y^{2}x+y}{x}dx=dy$. Simplify this further: $(y^2+\frac{y}{x})dx=dy$.

Integrate Both Sides

Integrate both sides of the equation. First, integrate the left side with respect to $x$: $\int (y^2+\frac{y}{x})dx$. The right side will be integrated with respect to $y$: $\int dy$.

Solve the Integrals

Solve the integrals: $$\int y^{2}dx+\int \frac{y}{x}dx=y{x}^2+y\ln |x|=F(y)+Cdy$$. Integrate the terms for the respective variables. Hence: $y{x}^2+y\ln |x|=y+C$.

Identification of the Correct Solution

Compare the integrated format to the given options. The result that satisfies all conditions of the solved integral is: Option (3): $$\frac{x^2}{2}+\frac{x}{y}+c=0$$.

Key concepts

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Variable separation is a fundamental technique to solve differential equations. It involves rearranging the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the other side. This allows us to integrate both sides easily.
For example, consider the differential equation given in the problem: $y^{2}xdx+ydx-xdy=0$. To separate the variables, we first combine terms involving $dx$ and $dy$: $(y^{2}x+y)dx=xdy$.
Next, we rearrange terms to isolate variables on either side: $\frac{(y^{2}x+y)}{x}dx=dy$. Simplify further for clarity: $(y^2+\frac{y}{x})dx=dy$. This separation helps us move towards integration of each side with respect to its respective variable.

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Integration is the process of finding an integral, which represents the area under a curve within a given interval. It is a crucial step in solving differential equations once variables are separated.
Given our separated differential equation: $(y^2+\frac{y}{x})dx=dy$, we need to integrate both sides.
First, integrate the left side with respect to $x$: $\mathrm{\setminus }\int (y^2+\frac{y}{x})dx$.
Next, integrate the right side with respect to $y$: $\mathrm{\setminus }\int dy$.
The integrals are solved as follows:
$\int y^{2}dx+\int \frac{y}{x}dx=y{x}^2+y\mathrm{\setminus }ln|x|=F(y)+C\mathrm{\setminus }dy$.
Simplifying these integrals provides the relationship: $y{x}^2+y\mathrm{\setminus }ln|x|=y+C$. This leads to the solution format given in the options.

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IIT JEE preparation demands a deep understanding of key concepts like differential equations. Mastery of methods like variable separation and integration is essential, as these are common in the exam.
Steps to Solving Differential Equations for IIT JEE:

• Read the problem carefully: Understand what is given and what is required.
• Rearrange the equation: Combine and rearrange terms to isolate variables on each side.
• Separate the variables: Make sure each side of the equation depends on a single variable for easy integration.
• Integrate both sides: Perform the integration considering proper variable limits or boundaries if provided.

Finally, compare your solution with the given options, ensuring all steps are verified and any constants of integration are appropriately handled.
Utilizing these steps methodically allows for a systematic approach in tackling similar problems effectively during your IIT JEE preparation.