3.1E: Exercises

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) \(y''y-(y')^2=2\)

Answer

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

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

Answer

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

Answer

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

Answer

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

Answer

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

Answer

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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.

Answer

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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)?

Answer

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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 \(x=0\)

Answer

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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).

Answer

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