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Chapter 5: Problem 2

Solve the differential equation. $$ \frac{d y}{d x}=4-y $$

Short Answer

Expert verified
The solution to the given differential equation is \(y = 4 卤 C1e^{-x}\).

Step by step solution

01

Rearrange The Equation

Rearrange the differential equation in order to separate the variables. This is done by moving all terms involving \(y\) to one side and all terms involving \(x\) to the other.\nSo, we re-write the differential equation as, \(dy/(4 - y) = dx.\)
02

Integrate Both Sides

Once the equation is rearranged, the next step is to integrate both sides. When integrating, remember to include the constant of integration 'C'.\nThus, we obtain, \(\int dy/(4 - y) = \int dx.\) This will result in \(- ln|4-y| = x + C. \)
03

Solve For Y

To isolate \(y\), first, remove the natural logarithm using an exponential function. This will give |4-y| = e^(-x-C). Then, write this absolute value equation as a piecewise function and solve for \(y\).\nSo, finally we get, \(y = 4 - e^{C}e^{-x}\) or \(y = 4 + e^{C}e^{-x}\). Note that \(e^{C}\) is just another constant, we can denote it as \(C1\). Therefore, the general solution to the given differential equation is \(y = 4 卤 C1e^{-x}\).

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Most popular questions from this chapter

The area of the region bounded by the graphs of \(y=x^{3}\) and \(y=x\) cannot be found by the single integral \(\int_{-1}^{1}\left(x^{3}-x\right) d x\). Explain why this is so. Use symmetry to write a single integral that does represent the area.

Find the length of the curve \(y^{2}=x^{3}\) from the origin to the point where the tangent makes an angle of \(45^{\circ}\) with the \(x\) -axis.

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=\arctan x, \quad y=0, \quad x=0, \quad x=1\) (a) 10 (b) \(\frac{3}{4}\) (c) 5 (d) -6 (e) 15

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ f(x)=x^{4}-4 x^{2}, g(x)=x^{3}-4 x $$

In Exercises \(53-56,\) use integration to find the area of the figure having the given vertices. $$ (2,-3),(4,6),(6,1) $$
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