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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 differential equation of all conics having centre at the origin is of order. (A) 2 (B) 3 (C) 4 (D) 5

The general solution of \([\mathrm{x}(\mathrm{dy} / \mathrm{dx})-\mathrm{y}] \mathrm{e}^{(\mathrm{y} / \mathrm{x})}=\mathrm{x}^{2} \cos \mathrm{x}\) is: (A) \(\mathrm{e}^{(\mathrm{x} / \mathrm{y})}=\cos \mathrm{x}+\mathrm{c}\) (B) \(\mathrm{e}^{(\mathrm{x} / \mathrm{y})}=\sin \mathrm{x}+\mathrm{c}\) (C) \(e^{(y / x)}=\sin x+c\) (D) \(e^{(y / x)}=\cos x+c\)

If \(\mathrm{f}(\mathrm{x})\) and \(\mathrm{g}(\mathrm{x})\) are two solutions of the differential equation \(\mathrm{q}\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)+\mathrm{x}^{2}(\mathrm{dy} / \mathrm{d} \mathrm{x})+\mathrm{y}=\mathrm{e}^{\mathrm{x}}\), then \(\mathrm{f}(\mathrm{x})-\mathrm{g}(\mathrm{x})\) is the solution of: (A) \(\mathrm{q}\left(\mathrm{d}^{2} \mathrm{y} / \mathrm{d} \mathrm{x}^{2}\right)+\mathrm{y}=\mathrm{e}^{\mathrm{x}}\) (B) \(q^{2}\left(d^{2} y / d x^{2}\right)+(d y / d x)+y=e^{x}\) (C) \(q^{2}\left(d^{2} y / d x^{2}\right)+y=e^{x}\) (D) \(q\left(d^{2} y / d x^{2}\right)+x^{2}(d y / d x)+y=0\)

The differential equation which represents the family of curves \(\mathrm{y}=\mathrm{c}_{1} \mathrm{e}^{(\mathrm{c}) 2 \mathrm{x}}\), where \(\mathrm{c}_{1}\) and \(\mathrm{c}_{2}\) are arbitarary constants, is: (A) \(y^{\prime}=y^{2}\) (B) \(\mathrm{y}^{\prime \prime}=\mathrm{y}^{\prime} \mathrm{y}\) (C) \(\mathrm{yy}^{\prime \prime}=\left(\mathrm{y}^{1}\right)^{2}\) (D) \(\mathrm{yy}^{\prime \prime}=\mathrm{y}^{\prime}\)

If \(\sin (x+y)(d y / d x)=5\) then (A) \(5 \int[(\mathrm{dt}) /(5+\sin \mathrm{t})]=\mathrm{t}+\mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (B) \(5 \int[(\mathrm{dt}) /(5+\sin \mathrm{t})]=\mathrm{t}-\mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (C) \(\int[(\mathrm{dt}) /(5+\operatorname{cosec} \mathrm{t})]=\mathrm{d} \mathrm{x}(\) where \(\mathrm{t}=\mathrm{x}+\mathrm{y})\) (D) \(\int[(\mathrm{dt}) /(5 \sin t+1)]=\mathrm{dt}(\) where \(t=x+y)\)
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