Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

The solution of the equation \(\frac{d y}{d x}+x=x e^{(n-1) y}\) is (A) \(\frac{1}{(n-1)} \log \left(\frac{e^{(n-1) y}-1}{e^{(n-1) y}}\right)=\frac{x^{2}}{2}+c\) (B) \(e^{(n-1) y}=c e^{(n-1) y+(n-1)^{\frac{x^{2}}{2}}}+1\) (C) \(\log \left(\frac{e^{(n-1) y}-1}{(n-1) e^{(n-1) y}}\right)=n^{2}+c\) (D) \(e^{(n-1) y}=c e^{(n-1) \frac{x^{2}}{2}+x}+1\)

Step-by-Step Solution

Answer

The correct solution is (D): \(e^{(n-1)y} = ce^{(n-1)\frac{x^2}{2} + x} + 1\).

Step 1: Identify the Equation Type

The given differential equation is \(\frac{d y}{d x}+x=x e^{(n-1) y}\). This is in the form of a Bernoulli's differential equation. These equations can be simplified using a substitution method.

Step 2: Apply Substitution Method

For Bernoulli's equation \(\frac{dy}{dx} + P(x)y = Q(x)y^n\), if \(n = 1\), the equation can be linearized by the substitution \(v = y^{1-n}\). Here, \(n-1\) acts like \(n\) in the standard form of Bernoulli's equation.

Step 3: Perform the Substitution

With the substitution \(v = e^{(n-1)y}\), it implies that \(\frac{dv}{dx} = (n-1)e^{(n-1)y}\frac{dy}{dx}\). Substituting back into the original equation gives another form: \(\frac{1}{n-1}\frac{dv}{dx} = xe^{(n-1)y} - x\).

Step 4: Simplify the Equation

Substitute back \(e^{(n-1)y} = v\) to rewrite the equation: \((n-1)\frac{dv}{dx} = xv - x\). This simplifies the problem, allowing further integration.

Step 5: Rearrange and Integrate

Rearrange the equation to isolate \(x\) and \(v\): \(\frac{dv}{v-1} = x\,dx\). Integrating both sides, \(\int \frac{dv}{v-1} = \int x\,dx\). The integration yields: \(\ln|v-1| = \frac{x^2}{2} + C\), where \(C\) is the integration constant.

Step 6: Solve for Original Variable

Recall that \(v = e^{(n-1)y}\). Substitute back \(v\) and solve for \(e^{(n-1)y}\): \(e^{(n-1)y} = C e^{\frac{x^2}{2}} + 1\).

Step 7: Verify Solution in Original Equation

Finally, match this result against the options given to confirm which expression it matches exactly. The correct option is (D): \(e^{(n-1)y} = ce^{(n-1)\frac{x^2}{2} + x} + 1\).

Key Concepts

Differential Equation

Differential equations are mathematical equations that relate a function to its derivatives. These equations are essential in describing various phenomena such as motion, heat, and waves in physics. In this context, we are focusing on a specific type of differential equation known as Bernoulli's differential equation.
This is identified by its structure, which often takes the form: \( \frac{dy}{dx} + P(x)y = Q(x)y^n \). When you see an equation like this, it signals the need to simplify using a unique approach. Understanding the basics of a differential equation involves recognizing how changes in variables can affect the system being modeled. Always look for opportunities to transform complicated expressions into simpler linear forms.

• Note the involvement of both the function \( y \) and its derivative \( \frac{dy}{dx} \) in the equation.
• These equations can model dynamic processes, providing insights into rates of change within those systems.
• Bernoulli's form is conducive to techniques involving substitution, as it allows simplification even when terms are nonlinear.

Integration

Integration is a fundamental concept in calculus that allows us to find functions given their derivatives, essentially working backwards from differentiation. In our problem, integration plays a central role in finding the solution after simplifying the differential equation.
After substitution and rearrangement in Bernoulli's equation, the integration of both sides yields insights into the solution. When integrating, one often encounters a constant of integration, often denoted as \( C \). This constant is critical in ensuring the general solution accounts for all potential initial conditions.

• Indefinite integral represents a family of functions, reflecting all possible scenarios
• The definite integral would represent a specific solution with given limits
• Recognizing common integrals can speed up this process, making it more approachable

Ultimately, through integration, we derive the equation \( \ln|v-1| = \frac{x^2}{2} + C \), which expands the solution set for our problem context.

Substitution Method

The substitution method in differential equations serves to simplify complex expressions, often transforming nonlinear equations into a more tractable linear form. By recognizing a substitute variable or expression, you reduce one of the equation's complexities.
In the case of Bernoulli's differential equation, employing the substitution \( v = e^{(n-1)y} \) simplifies the problem magnificently. This transformation switches the focus from the original variable \( y \) to a new working variable \( v \). Through substitution:

• Focus is shifted from the original equation to a simpler counterpart
• Presents a pathway to linearize nonlinear components
• Reduces the order of the equation, simplifying potential integrations and manipulations

Such a method is not confined to Bernoulli's equations alone. It's a powerful strategy across various domains in calculus, allowing intricate problems to unfold into understandable patterns. Successfully maneuvering through substitution often delineates the difference between grappling with a problem and achieving its elegant solution.