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# 3.1E: Exercises

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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## Exercises

### Exercise $$\PageIndex{1}$$

Find the order of the equation.

(a) $$\displaystyle{d^2y\over dx^2}+2{dy\over dx}\ {d^3y\over dx^3}+x=0$$

(b) $$y''-3y'+2y=x^7$$

(c) $$y'-y^7=0$$

(d) $$y''y-(y')^2=2$$

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### Exercise $$\PageIndex{2}$$

Verify that the function is a solution of the differential equation on some interval, for any choice of the arbitrary constants appearing in the function.

(a) $$y=ce^{2x}; \quad y'=2y$$

(b) $$y={x^2\over3}+{c\over x}; \quad xy'+y=x^2$$

(c) $$y={1\over2}+ce^{-x^2}; \quad y'+2xy=x$$

(d) $$y=(1+ce^{-x^2/2}); (1-ce^{-x^2/2})^{-1} \quad 2y'+x(y^2-1)=0$$

(e) $$y={\tan\left( {x^3\over3}+c\right)}; \quad y'=x^2(1+y^2)$$

(f) $$y=(c_1+c_2x)e^x+\sin x+x^2; \quad y''-2y'+y=-2 \cos x+x^2-4x+2$$

(g) $$y=c_1e^x+c_2x+{2\over x}; \quad (1-x)y''+xy'- y=4(1-x-x^2)x^{-3}$$

(h) $$y=x^{-1/2}(c_1\sin x+c_2 \cos x)+4x+8$$;

$$x^2y''+xy'+{\left(x^2-{1\over4}\right)}y=4x^3+8x^2+3x-2$$

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### Exercise $$\PageIndex{3}$$

Find all solutions of the equation.

(a) $$y'=-x$$

(b) $$y'=-x \sin x$$

(c) $$y'=x \ln x$$

(d) $$y''=x \cos x$$

(e) $$y''=2xe^x$$

(f) $$y''=2x+\sin x+e^x$$

(g) $$y'''=-\cos x$$

(h) $$y'''=-x^2+e^x$$

(i) $$y'''=7e^{4x}$$

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### Exercise $$\PageIndex{4}$$

Solve the initial value problem.

(a) $$y'=-xe^x, \quad y(0)=1$$

(b) $$y'=x \sin x^2, \quad y\left({\sqrt{\pi\over2}}\right)=1$$

(c) $$y'=\tan x, \quad y(\pi/4)=3$$

(d) $$y''=x^4, \quad y(2)=-1, \quad y'(2)=-1$$

(e) $$y''=xe^{2x}, \quad y(0)=7, \quad y'(0)=1$$

(f) $$y''=- x \sin x, \quad y(0)=1, \quad y'(0)=-3$$

(g) $$y'''=x^2e^x, \quad y(0)=1, \quad y'(0)=-2, \quad y''(0)=3$$

(h) $$y'''=2+\sin 2x, \quad y(0)=1, \quad y'(0)=-6, \quad y''(0)=3$$

(i) $$y'''=2x+1, \quad y(2)=1, \quad y'(2)=-4, \quad y''(2)=7$$

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### Exercise $$\PageIndex{5}$$

Verify that the function is a solution of the initial value problem.

(a) $$y=x\cos x; \quad y'=\cos x-y\tan x, \quad y(\pi/4)={\pi\over4\sqrt{2}}$$

(b) $${y={1+2\ln x\over x^2}+{1\over2}; \quad y'={x^2-2x^2y+2\over x^3}, \quad y(1)={3\over2}}$$

(c) $$y={\tan\left({x^2\over2}\right)}; \quad y'=x(1+y^2), \quad y(0)=0$$

(d) $${y={2\over x-2}; \quad y'={-y(y+1)\over x}}, \quad y(1)=-2$$

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### Exercise $$\PageIndex{6}$$

Verify that the function is a solution of the initial value problem.

(a) $$y=x^2(1+\ln x); \quad y''={3xy'-4y\over x^2}, \quad y(e)=2e^2, \quad y'(e)=5e$$

(b) $$y={x^2\over3}+x-1; \quad y''={x^2-xy'+y+1\over x^2}, \quad y(1)={1\over3}, \quad y'(1)={5\over3}$$

(c) $$y=(1+x^2)^{-1/2}; \quad y''={(x^2-1)y-x(x^2+1)y'\over (x^2+1)^2}, \quad y(0)=1, y'(0)=0$$

(d) $$y={x^2\over 1-x}; \quad y''={2(x+y)(xy'-y)\over x^3}, \quad y(1/2)=1/2, \quad y'(1/2)=3$$

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### Exercise $$\PageIndex{7}$$

Suppose an object is launched from a point 320 feet above the earth with an initial velocity of 128 ft/sec upward, and the only force acting on it thereafter is gravity. Take $$g=32 ft/sec^2$$

(a) Find the highest altitude attained by the object.

(b) Determine how long it takes for the object to fall to the ground.

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### Exercise $$\PageIndex{8}$$

Let $$a$$ be a nonzero real number.

(a) Verify that if $$c$$ is an arbitrary constant then equation A: $$y=(x-c)^a$$ is a solution of equation B: $$y'=ay^{(a-1)/a}$$ on $$(c,\infty)$$.

(b) Suppose $$a<0$$ or $$a>1$$. Can you think of a solution of (B) that isn't of the form (A)?

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### Exercise $$\PageIndex{9}$$

Verify that $$y= e^x-1, x \ge 0$$ and $$1-e^{-x}, x < 0,$$ is a solution of $$y'=|y|+1$$ on $$(-\infty,\infty)$$.

Hint

Use the definition of derivative at $$x=0$$

### Exercise $$\PageIndex{10}$$
(a) Verify that if $$c$$ is any real number then equation A: $$y=c^2+cx+2c+1$$ satisfies equation B: $$y'={-(x+2)+\sqrt{x^2+4x+4y}\over2}$$ on some open interval. Identify the open interval.
(b) Verify that $$y_1={-x(x+4)\over4}$$ also satisfies (B) on some open interval, and identify the open interval. (Note that $$y_1$$ can't be obtained by selecting a value of $$c$$ in (A).