Question

Let y = y₁(x) and y = y₂(x) be two distinct solution of the differential equation
\(\frac{dy}{dx} = x+y,\)
with y₁(0) = 0 and y₂(0) = 1 respectively. Then, the number of points of intersection of y = y₁ (x) and y = y₂(x) is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The Correct Option is A

To determine the number of points of intersection of the two functions \(y = y_1(x)\) and \(y = y_2(x)\), we first need to solve the differential equation given:

\[ \frac{dy}{dx} = x + y \]

This is a first-order linear differential equation, which can be solved using the integrating factor method. The standard form of this differential equation is given by:

\[ \frac{dy}{dx} - y = x \]

Here, the integrating factor \(I(x)\) is calculated as:

\[ I(x) = e^{\int -1 \, dx} = e^{-x} \]

Multiplying the entire differential equation by the integrating factor, we have:

\[ e^{-x} \frac{dy}{dx} - e^{-x}y = e^{-x}x \]

The left-hand side of the equation is a derivative of \(y e^{-x}\), so we can write:

\[ \frac{d}{dx}(ye^{-x}) = xe^{-x} \]

Integrating both sides with respect to \(x\), we obtain:

\[ ye^{-x} = \int x e^{-x} \, dx \]

The integral on the right side, \(\int x e^{-x} \, dx\), can be solved using integration by parts. Let \(u = x\) and \(dv = e^{-x} \, dx\). Then \(du = dx\) and \(v = -e^{-x}\). Applying integration by parts:

\[ \int x e^{-x} \, dx = -xe^{-x} + \int e^{-x} \, dx = -xe^{-x} - e^{-x} + C \]

Thus, we have:

\[ ye^{-x} = -xe^{-x} - e^{-x} + C \]

Multiply through by \(e^x\):

\[ y = -x - 1 + Ce^x \]

Using the initial conditions, we determine the constants for \(y_1\) and \(y_2\).

• For \(y_1(x)\) with \(y_1(0) = 0\):

\[ 0 = -0 - 1 + C \implies C = 1 \]

Therefore, \(y_1 = -x - 1 + e^x\).

• For \(y_2(x)\) with \(y_2(0) = 1\):

\[ 1 = -0 - 1 + C \implies C = 2 \]

Therefore, \(y_2 = -x - 1 + 2e^x\).

Now, we'll find the intersection points by equating \(y_1(x)\) and \(y_2(x)\):

\[ -x - 1 + e^x = -x - 1 + 2e^x \]

Simplifying, we get:

\[ e^x = 2e^x \]

Subtracting \(e^x\) from both sides, we get:

\[ 0 = e^x \]

Since \(e^x\) is never zero for any real \(x\), there are no solutions to this equation. Thus, \(y_1(x)\) and \(y_2(x)\) do not intersect.

Conclusion: The number of points of intersection between the two solutions is 0.