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Chapter 13: Problem 1177

The solution of \((\mathrm{dy} / \mathrm{dx})=4 \mathrm{x}+\mathrm{y}+1 \mathrm{is}:\) (A) \(4 \mathrm{x}+\mathrm{y}+1=\mathrm{c} \cdot \mathrm{e}^{\mathrm{x}}\) (B) \(4 \mathrm{x}+\mathrm{y}+5=\mathrm{e}^{\mathrm{x}}+\mathrm{c}\) (C) \(4 \mathrm{x}+\mathrm{y}+5=\mathrm{c} \cdot \mathrm{e}^{\mathrm{x}}\) (D) none of these

Short Answer

Expert verified
The short answer based on the step-by-step solution is: \[4x+y+5=Ce^x.\] The correct choice is (C).

Step by step solution

01

Identify the Given Differential Equation

We are given the first-order linear differential equation: \(\frac{\mathrm{dy}}{\mathrm{dx}} = 4x + y + 1\).
02

Write the Standard Form of a First-Order Linear Differential Equation

The general form of a first-order linear differential equation is: \(\frac{\mathrm{dy}}{\mathrm{dx}} + P(x)y = Q(x)\).
03

Identify P(x) and Q(x)

Comparing the given equation with the standard form, we get: \(P(x) = -1\) and \(Q(x) = 4x + 1\).
04

Calculate the Integrating Factor

The integrating factor is given by: \(I(x) = e^{\int P(x) \,\mathrm{d}x}\). Substituting for P(x), we get: \(I(x) = e^{\int (-1) \,\mathrm{d}x}\) which simplifies to: \(I(x) = e^{-x}\).
05

Multiply the Given Equation by the Integrating Factor

Multiply both sides of the equation by \(I(x)\) to get: \(\frac{\mathrm{dy}}{\mathrm{dx}} e^{-x} - ye^{-x} = (4x + 1)e^{-x}\).
06

Integrate Both Sides

Upon integrating both sides, we get: \[\int \left( \frac{\mathrm{d}y}{\mathrm{d}x} e^{-x} - ye^{-x} \right)\mathrm{d}x = \int (4x + 1)e^{-x}\,\mathrm{d}x.\]
07

Solve for y

Integrate the right side using integration by parts. The left side becomes a product of derivatives which simplifies to: \(y(x)e^{-x} = -4(x+1)e^{-x} - Ce^{-x}\), where C is the integration constant. Multiply by \(e^x\) to get: \(y(x)= -4(x+1)-Ce^x\). The general solution of the given differential equation has the form: \(y(x)=-4(x+1)-Ce^x\). Comparing this to the given options to identify the correct choice: (A) \(4x + y + 1 = Ce^x\) is incorrect. (B) \(4x + y + 5 = e^x + C\) is incorrect. (C) \(4x + y + 5 = Ce^x\) is correct. So, the answer is (C). (D) None of these would be incorrect, since we found a match in option (C).

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Most popular questions from this chapter

The solution of \(\mathrm{xdy}-\mathrm{ydx}=0\) represents: (A) parabola having vertex at \((0,0)\) (B) circle having centre at \((0,0)\) (C) a straight line passing through \((0,0)\) (D) a rectangular hyperbola

\(\mathrm{y}^{2}=(\mathrm{x}-\mathrm{c})^{3}\) is general solution of the differential equation: (where c is arbitrary constant). (A) \((\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (B) \(2(\mathrm{dy} / \mathrm{dx})^{3}-8 \mathrm{y}=0\) (C) \(8(\mathrm{dy} / \mathrm{dx})^{3}=27 \mathrm{y}\) (D) \(8\left(\mathrm{~d}^{3} \mathrm{y} / \mathrm{dx}^{3}\right)-27 \mathrm{y}=0\)

The solution of the equation \(\mathrm{x}+\mathrm{y}(\mathrm{dy} / \mathrm{d} \mathrm{x})=2 \mathrm{y}\) is: (A) \(x y^{2}=c^{2}(x+2 y)\) (B) \(\mathrm{y}^{2}=\mathrm{c}\left(\mathrm{x}^{2}+2 \mathrm{y}\right)\) (C) \(\log (\mathrm{y}-\mathrm{x})=\mathrm{c}+[\mathrm{x} /(\mathrm{y}-\mathrm{x})]\) (D) \(\log [\mathrm{x} /(\mathrm{x}-\mathrm{y})]=\mathrm{c}+\mathrm{y}-\mathrm{x}\)

The equation of a curve passing through \([2,(7 / 2)]\) and having gradient \(1-\left(1 / \mathrm{x}^{2}\right)\) at \((\mathrm{x}, \mathrm{y})\) is: (A) \(x y=x+1\) (B) \(x^{2}+x+1\) (C) \(x y=x^{2}+x+1\) (D) none of these

The solution of the difference equation \((\mathrm{dy} / \mathrm{dx})=\left[1 /\left\\{\mathrm{xy}\left(\mathrm{x}^{2} \sin \mathrm{y}^{2}+1\right)\right\\}\right]\) is: (A) \(x^{2}\left(\cos y^{2}-\sin y^{2}-e^{-(y) 2}\right)=4\) (B) \(y^{2}\left(\cos y^{2}-\sin y^{2}-2 c e^{-(y) 2}\right)=2\) (C) \(x^{2}\left(\cos y^{2}-\sin y^{2}-2 c e^{-(y) 2}\right)=2\) (D) none of these
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