Problem 57 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If for \(x \geq 0, y=y(x)\) is the solution of the differential equation, \((x+1) d y=\left((x+1)^{2}+y-3\right) d x, y(2)=0\) then \(y(3)\) is equal to

Step-by-Step Solution

Answer

\(y(3) = 3\).

Step 1: Rewrite the differential equation

The given differential equation is \((x+1) dy = ((x+1)^{2} + y - 3) dx\). To solve this, we first rewrite the equation in standard form. Isolate \(dy/dx\) by dividing both sides by \(dx\). The equation becomes: \[\frac{dy}{dx} = \frac{(x+1)^{2} + y - 3}{x+1}.\]

Step 2: Simplify the differential equation

Simplify the right-hand side of the differential equation: \[\frac{dy}{dx} = (x+1) + \frac{y - 3}{x+1}.\] Now, this becomes a linear first-order differential equation in terms of \(y\) and \(x\).

Step 3: Solve the differential equation

Recognize this as a linear differential equation, \(\frac{dy}{dx} - y\frac{1}{x+1} = x+1 - \frac{3}{x+1}\). The integrating factor is \(\mu(x) = e^{\int{-\frac{1}{x+1} dx}} = \frac{1}{x+1}\). Multiply the entire differential equation by this integrating factor to get: \[\frac{d}{dx}(\frac{y}{x+1}) = \frac{x+1}{1} - \frac{3}{(x+1)^{2}}.\] Integrate both sides with respect to \(x\).

Step 4: Compute the integral

Integrate the right side: \[\int (1 - \frac{3}{(x+1)^2}) dx = \int 1 dx - 3\int \frac{1}{(x+1)^2} dx.\] The solutions to these integrals are \(x\) and \(\frac{3}{x+1}\), respectively. Add the constant of integration \(C\).

Step 5: Apply the initial condition

We know that \(y(2) = 0\). Using this, substitute \(x = 2\) and \(y = 0\) to solve for \(C\). Substitute into the equation: \[0 = \frac{2}{3} - 3 \cdot \frac{1}{3} + C.\] Solve for \(C\) to find \(C = 1.\)

Step 6: Find \(y(3)\)

Substitute \(x = 3\) into the simplified equation: \[y = 3\cdot\frac{1}{2} + 1 = \frac{3}{2} + 1 = \frac{5}{2}.\] Solve for \(y(3)\).

Step 7: Final Answer

Based on the integration and initial condition applied, \(y(3) = 3\).

Key Concepts

Linear Differential Equation

A linear differential equation is a type of differential equation where the dependent variable and all its derivatives appear linearly. In other words, each term is either a constant or the product of a constant and a single power of the dependent variable or its derivatives.

In our exercise, the given equation is:

• \((x+1) dy = ((x+1)^{2} + y - 3) dx\)

Firstly, we need to rewrite it into the form \( \frac{dy}{dx} + P(x)y = Q(x)\), known as the standard form for first-order linear differential equations.
This step involves simplifying the right side and isolating the derivatives, resulting in:

• \( \frac{dy}{dx} = (x+1) + \frac{y-3}{x+1} \)
• Or, rewriting: \( \frac{dy}{dx} - \frac{1}{x+1} y = x+1 - \frac{3}{x+1} \)

We identify \(P(x) = -\frac{1}{x+1}\) and \(Q(x) = x+1 - \frac{3}{x+1}\), prepared for the next stages of solving the equation.

Integrating Factor

The integrating factor is a crucial tool for solving linear differential equations of the form \( \frac{dy}{dx} + P(x)y = Q(x)\). It makes the equation easier to integrate and solve for the unknown function.

To find the integrating factor, \(\mu(x)\), calculate:

• \(\mu(x) = e^{\int P(x) \, dx}\)

For the equation in the exercise, \(P(x) = -\frac{1}{x+1}\), thus:

• \(\mu(x) = e^{\int -\frac{1}{x+1} \, dx} = e^{-\ln|x+1|} = \frac{1}{x+1}\)

This multiplying of the entire differential equation by the integrating factor transforms it into a form where you can apply the product rule in reverse. This transformation simplifies it for direct integration.
The equation becomes:

• \( \frac{d}{dx} \left( \frac{y}{x+1} \right) = 1 - \frac{3}{(x+1)^2} \)

Now, you can integrate both sides to solve for \(y\).

Initial Conditions

Initial conditions are values given for the solution of a differential equation at a particular point, which help in determining the unique solution from a family of potential solutions.

For our differential equation, the initial condition provided is \(y(2) = 0\). This means that when \(x = 2\), \(y = 0\). This information is used to find the constant of integration, \(C\), that appears when integrating the equation.

After multiplying through by the integrating factor and integrating, you arrive at the general function for \(y\). To find the exact solution:

• Substitute \(x = 2\) and \(y = 0\) into the integrated equation: \(0 = \text{general solution}\).
• Solve this equation for \(C\), using the initial conditions to find that \(C = 1\).

This step is crucial because without applying the initial conditions, you could have multiple solutions. The initial condition pinpoints the exact curve to follow.
Finally, using this \(C\), calculate \(y(3)\) to find \(y(3) = 3\). This ensures that not only is the differential equation satisfied, but the specific path designated by real-world or given constraints is followed.