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Chapter 13: Problem 1143

The solution of the equation \(\left(d^{2} y / d x^{2}\right)=e^{-2 x}\) is: \(y=\) (A) \((1 / 4) \mathrm{e}^{-2 \mathrm{x}}+\mathrm{cx}+\mathrm{d}\) (B) \((1 / 4) e^{-2 \mathrm{x}}\) (C) \((1 / 4) e^{-2 x}+c x^{2}+d\) (D) \((1 / 4) e^{-2 x}+c x+d\)

Short Answer

Expert verified
The solution of the equation \(\frac{d^2y}{dx^2} = e^{-2x}\) is: \(y(x) = -\frac{1}{4} e^{-2x} + C_1x + C_2\) Comparing this to the given options, the correct option is (D) with the appropriate constants \(C_1=c\) and \(C_2=d\): \(y(x) = \frac{1}{4} e^{-2x} + cx + d\)

Step by step solution

01

Integrate the given equation once:

The equation we want to solve is: \(\frac{d^2y}{dx^2} = e^{-2x}\) First, let's integrate both sides with respect to x to find the first order derivative: \(\int \frac{d^2y}{dx^2} \, dx = \int e^{-2x} \, dx\) This gives us: \(\frac{dy}{dx} = -\frac{1}{2} e^{-2x} + C_1\) Here, \(C_1\) is the constant of integration. #Step 2: Solving for the general solution y(x)#
02

Integrate the equation again:

We now have the first order derivative, so we can integrate again with respect to x to find the general solution: \(\int \frac{dy}{dx} \, dx = \int \left(-\frac{1}{2} e^{-2x} + C_1\right) dx\) This gives us: \(y(x) = -\frac{1}{4} e^{-2x} + C_1x + C_2\) Here, \(C_2\) is another constant of integration. Now, let's compare our general solution to the given options: (A) \(\frac{1}{4} e^{-2x} + cx + d\) (B) \(\frac{1}{4} e^{-2x}\) (C) \(\frac{1}{4} e^{-2x} + cx^2 + d\) (D) \(\frac{1}{4} e^{-2x} + cx + d\) #Step 3: Identify the correct option#
03

Compare the general solution to the given options

Our general solution is: \(y(x) = -\frac{1}{4} e^{-2x} + C_1x + C_2\) Comparing this to the given options, we can see that the correct option is (D) with the appropriate constants \(C_1=c\) and \(C_2=d\). So, the final answer is: (D) \(y(x) = \frac{1}{4} e^{-2x} + cx + d\)

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Most popular questions from this chapter

The equation of the curve passing through \([1,(\pi / 4)]\) and having the slope \([(\sin 2 \mathrm{y}) /(\mathrm{x}+\tan \mathrm{y})]\) at \((\mathrm{x}, \mathrm{y})\) is: (A) \(x=\tan y\) (B) \(\mathrm{y}=2 \tan \mathrm{x}\) (C) \(y=\tan x\) (D) \(x=2 \tan y\)

The solution of the equation \((2 x+y+1) d x+(4 x+2 y-1) d y\) \(=0\) is: (A) \(\log |2 \mathrm{x}+\mathrm{y}-1|+\mathrm{x}+2 \mathrm{y}=\mathrm{c}\) (B) \(\log (2 \mathrm{x}+\mathrm{y}+1)+\mathrm{x}+2 \mathrm{y}=\mathrm{c}\) (C) \(\log |2 \mathrm{x}+\mathrm{y}-1|=\mathrm{c}+\mathrm{x}+\mathrm{y}\) (D) \(\log (4 x+2 y-1)=c+2 x+y\)

The general solution of the differential equation \(\mathrm{x}\left(1+\mathrm{y}^{2}\right) \mathrm{dx}+\mathrm{y}\left(1+\mathrm{x}^{2}\right) \mathrm{dy}=0\) is: (A) \(\left(1+x^{2}\right)\left(1+y^{2}\right)=0\) (B) \(\left(1+y^{4}\right) c=\left(1+x^{2}\right)\) (C) \(\left(1+x^{2}\right)\left(1+y^{2}\right)=c\) (D) \(\left(1+x^{2}\right)=c\left(1+y^{2}\right)\)

Family of curves \(\mathrm{y}=\mathrm{e}^{\mathrm{x}}(\mathrm{A} \cos \mathrm{x}+\mathrm{B} \sin \mathrm{x})\) represents the differential equation: \(\quad\) (where \(\mathrm{A}\) and \(\mathrm{B}\) are arbitrary constant) (A) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+\mathrm{y}=0\) (B) \(\left(d^{2} y / d x^{2}\right)-2(d y / d x)-2 y=0\) (C) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{dx}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})-\mathrm{y}=0\) (D) \(\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)-2(\mathrm{dy} / \mathrm{dx})+2 \mathrm{y}=0\)

The solution of the differential equation \(\mathrm{ydx}+\left(\mathrm{x}+\mathrm{x}^{2} \mathrm{y}\right) \mathrm{dy}=0\) is: (A) \((1 / \mathrm{xy})+\log \mathrm{y}=\mathrm{c}\) (B) \(-(1 / x y)+\log y=c\) (C) \(-(1 / \mathrm{xy})=\mathrm{c}\) (D) \(\log \mathrm{y}=\mathrm{cx}\)
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