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BITSAT 2010
Exam
BITSAT
Year
2010
Subject
Mathematics
Chapter
Differential Equations

BITSAT 2010 · Mathematics · Differential Equations

What is the solution of ((d y)/(d x)+2 y=1) satisfying y(0)=0 ?

What is the solution of (\frac{d y}{d x}+2 y=1) satisfying y(0)=0y(0)=0 ?

Options

  1. A(y=\frac{1-e^{-2 x}}{2})
  2. B(y=\frac{1+e^{-2 x}}{2})
  3. Cy=1+exy=1+e^x
  4. D(y=\frac{1+e^x}{2})

Correct answer

A. (y=\frac{1-e^{-2 x}}{2})

Step-by-step solution

(dydx+2y=1⇒dydx=1−2y∫dy1−2y=∫dx−12log⁡∣1−2y∣=x+C at x=0,y=0;−12log⁡1=0+C⇒C=01−2y=e−2x⇒y=1−e−2x2\begin{aligned} & \frac{d y}{d x}+2 y=1 \Rightarrow \frac{d y}{d x}=1-2 y \\ & \int \frac{d y}{1-2 y}=\int d x-\frac{1}{2} \log |1-2 y|=x+C \\ & \text{ at } x=0, y=0 ;-\frac{1}{2} \log 1=0+C \Rightarrow C=0 \\ & 1-2 y=e^{-2 x} \Rightarrow y=\frac{1-e^{-2 x}}{2} \end{aligned})
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