Let y_1(x) and y_2(x) be two solutions of the differential equation (d y)/(d x)=x . If y_1(0)=0 and y_2(0)=4 , then what is the number of points of intersection of curves y_1(x) and y_2(x) ?

Correct answer: A. No point Given, D.E. (d y)/(d x)=x aligned & ∫ d y & =∫ x d x & y & =(x^2)/(2)+c aligned y_1(x)=(x^2)/(2)+c_1 ...(i) and y_2(x)=(x^2)/(2)+c_2 ...(ii) For y_1(0)=0 in equation (i) 0=0+c_1 c_1=0 y_1(x)=(x^2)/(2) ...(iii) For y_2(0)=0 4=0+c_2 [ From (iii) ] For, c_2=4 y_2(x)=(x^2)/(2)+4 ...(iv) For number of points of intersection for y_1(x) and y_2(x) solving (iii) and (iv) (x^2)/(2)=(x^2)/(2)+4 (not possible) as 0 ≠ 4 Hence, no point of intersection.

NDA 2024

Exam: NDA
Year: 2024
Subject: Mathematics
Chapter: Differential Equations

NDA 2024 · Mathematics · Differential Equations

Let y_1(x) and y_2(x) be two solutions of the differential equation...

Let

y_1(x)

and

y_2(x)

be two solutions of the differential equation

\frac{d y}{d x}=x

. If

y_1(0)=0

and

y_2(0)=4

, then what is the number of points of intersection of curves

y_1(x)

and

y_2(x)

Options

ANo point
BOne point
CTwo points
DMore than two points

Correct answer

A. No point

Step-by-step solution

Given, D.E.

\frac{d y}{d x}=x

\begin{aligned} \Rightarrow & \int d y & =\int x d x \\ \Rightarrow & y & =\frac{x^2}{2}+c\end{aligned}

\Rightarrow \quad y_1(x)=\frac{x^2}{2}+c_1

...(i)

and \quad y_2(x)=\frac{x^2}{2}+c_2

...(ii) For

y_1(0)=0

in equation (i)

0=0+c_1\Rightarrow \quad c_1=0\Rightarrow \quad y_1(x)=\frac{x^2}{2} \quad

...(iii) For

\quad y_2(0)=0\Rightarrow \quad 4=0+c_2 \quad[

From (iii)

]

For,

\quad c_2=4\Rightarrow \quad y_2(x)=\frac{x^2}{2}+4 \quad

...(iv) For number of points of intersection for

y_1(x) and y_2(x)

solving (iii) and (iv)

\frac{x^2}{2}=\frac{x^2}{2}+4 \text{ (not possible) } as

0 \neq 4

Hence, no point of intersection.

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