1.2E: Basic Concepts (Exercises)

1. Find the order of the equation.

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

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

c. \(y'-y^7=0\)

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

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. \(y=x^{-1/2}(c_1\sin x+c_2 \cos x)+4x+8\); \(x^2y''+xy'+ {\left(x^2-{1\over4}\right)}y=4x^3+8x^2+3x-2\)

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

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

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

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

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\(^2\).

1. Find the highest altitude attained by the object.
2. Determine how long it takes for the object to fall to the ground.

8. Let \(a\) be a nonzero real number.

1. Verify that if \(c\) is an arbitrary constant then \[y=(x-c)^a \tag{A}\] is a solution of \[y'=ay^{(a-1)/a} \tag{B}\] on \((c,\infty)\).
2. Suppose \(a<0\) or \(a>1\). Can you think of a solution of (B) that isn’t of the form (A)?

9. Verify that \[\begin{aligned}y= \left\{ \begin{array}{cl} e^x-1,& x \ge 0, \\[6pt] 1-e^{-x},& x < 0, \end{array}\right.\end{aligned}\]

is a solution of

\[\begin{aligned}y'=|y|+1\end{aligned}\] on \((-\infty,\infty)\).

10.

(a) Verify that if \(c\) is any real number then \[y=c^2+cx+2c+1 \tag{A}\] satisfies \[y'={-(x+2)+\sqrt{x^2+4x+4y}\over2} \tag{B}\] on some open interval. Identify the open interval.

(b) Verify that \[\begin{aligned}y_1={-x(x+4)\over4}\end{aligned}\] 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).)