9.1E: Exercises for Section 9.1

In the following exercises, determine the order of each differential equation.

1) \(y′+y=3y^2\)

Answer

1st-order

2) \((y′)^2=y′+2y\)

Answer

1st-order

3) \(y'''+y''y′=3x^2\)

Answer

3rd-order

4) \(y′=y''+3t^2\)

Answer

2nd-order

5) \(\dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}=3x^4\)

Answer

2nd-order

6) \(\left(\dfrac{dy}{dt}\right)^2+8\dfrac{dy}{dt}+3y=4t\)

Answer

1st-order

In exercises 7-12, verify that the given function is a solution to the given differential equation.

7) \(y=2e^{−x}+x−1\quad\) solves \(\quad y′=x−y\)

8) \(y=e^{3x}−\dfrac{e^x}{2}\quad\) solves \(\quad y′=3y+e^x\)

9) \(y=2+x^{-3/2}\quad\) solves \(\quad 4x^2y''+12xy'+3y=6\)

10) \(y=e^x+\dfrac{\sin x}{2}−\dfrac{\cos x}{2}\quad\) solves \(\quad y′=\cos x+y\)

11) \(x^2y+y=4\quad\) solves \(\displaystyle \quad y′=-\frac{2xy}{x^2+2y}\)

Hint

Use implicit differentiation to find \(dy/dx\).

12) \(y=\sin(xy)\quad\) solves \(\displaystyle \quad y′=\frac{y \cos(xy)}{1-x\cos(xy)}\)

In the following exercises, verify the given general solution and find the particular solution.

13) Find the particular solution to the differential equation \(y′=3x^3\) that passes through \((1,4.75)\), given that \(y=C+\dfrac{3x^4}{4}\) is a general solution.

Answer

\(y=4+\dfrac{3x^4}{4}\)

14) Find the particular solution to the differential equation \(y′=2xy\) that passes through \(\left(0,\frac{1}{2}\right)\), given that \(y=Ce^{x^2}\) is a general solution.

Answer

\(y=\frac{1}{2}e^{x^2}\)

15) Find the particular solution to the differential equation \(y′x^2=y\) that passes through \(\left(1,\frac{2}{e}\right)\), given that \(y=Ce^{−1/x}\) is a general solution.

Answer

\(y=2e^{−1/x}\)

16) Find the particular solution to the differential equation \(\dfrac{du}{dt}=\tan u\) that passes through \(\left(1,\frac{π}{2}\right)\), given that \(u=\sin^{−1}\big(e^{C+t}\big)\) is a general solution.

Answer

\(u=\sin^{−1}\big(e^{−1+t}\big)\)

17) Find the particular solution to the differential equation \(y′(1−x^2)=1+y\) that passes through \((0,−2),\) given that \(y=C\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\) is a general solution.

Answer

\(y=−\dfrac{\sqrt{x+1}}{\sqrt{1−x}}−1\)

In the following exercises, find the general solution to the differential equation.

18) \(y′=\ln x+\tan x\)

Answer

\(y=C−x+x\ln x−\ln(\cos x)\)

19) \(y′=4^x +e^x - 2x\)

Answer

\(y=C+\dfrac{4^x}{\ln 4} + e^x + x^2\)

20) \(y′=\sin^{−1}(2x)\)

Answer

\(\displaystyle y=x \arcsin \! \left(2 x \right)+\frac{\sqrt{1-4 x^{2}}}{2} +C \)

21) \(y′=2t\sqrt{t^2+16}\)

Answer

\(y=\frac{2}{3}\sqrt{t^2+16}\big(t^2+16\big)+C\)

22) \(x′=t\sqrt{4+t}\)

Answer

\(x=\frac{2}{15}\sqrt{4+t}\big(3t^2+4t−32\big)+C\)

23) \(y′=\dfrac{y}{x}\)

Answer

\(y=Cx\)

In the following exercises, solve the initial-value problems. Separate variables and integrate both sides.

24) \(\dfrac{dy}{dt}=−t; \; y(0)=1. \)

Answer

\(y=1−\dfrac{t^2}{2},\)

25) \(\dfrac{dy}{dt}=−y; \; y(0)=-1.\)

Answer

\(y=−e^{−t}\)

26) \(\dfrac{dy}{dx}=2xy+y; \; y(0)=3\)

Answer

Hint: the right-hand side is \(y(2x+1)\). Answer is \(y=3e^{x^2+x}\).

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