9.4E: Exercises for Section 9.4

In exercises 1 - 5, state if each of the following differential equations is linear? Explain your reasoning.

1) \(x\dfrac{dy}{dx}=x^2y+2x^3 \sin x\)

Answer

Yes. The equation is in the form \(\displaystyle \frac{dy}{dx}+P(x)y=Q(x)\) where \(P(x)=-x\) and \(Q(x)=2x^2\sin x\).

2) \(\displaystyle \frac{dy}{dt}=ty\)

Answer

Yes.

3) \(\displaystyle \frac{dy}{dt}+y^2=x\)

Answer

No. Why?

4) \(\displaystyle y' + 1 = \frac{y}{x} + e^{x^2} \)

Answer

Yes.

5) \(y'=y+e^y\)

Answer

No.

In exercises 6 - 10, write the following first-order differential equations in standard form.

6) \(y'=x^3y+\sin x\)

Answer

\(y'−x^3y=\sin x\)

7) \(y'+3y−\ln x=0\)

Answer

\(y'-3y=\ln x\)

8) \(−xy'=(3x+2)y+xe^x\)

Answer

\(y'+\frac{(3x+2)}{x}y=−e^x\)

9) \(\dfrac{dy}{dt}=4y+ty+\tan t\)

Answer

\(\dfrac{dy}{dt}-(4+t)y = \tan t\)

10) \(\dfrac{dy}{dt}=yx(x+1)\)

Answer

\(\dfrac{dy}{dt}−yx(x+1)=0\)

In exercises 11 - 15, state the integrating factor for each of the following differential equations.

11) \(\displaystyle y'=xy+3\)

Answer

\(\displaystyle I=e^{-x^2/2}\)

12) \(\displaystyle y' = \frac{y}{2x} + \sin x\)

Answer

\(\displaystyle I = \frac{1}{\sqrt{x}}\)

13) \(\displaystyle y'+ \frac{4y}{x+2} = 3x \)

Answer

\(I=(x+2)^4\)

14) \(\dfrac{dy}{dx}+y\tan x =\cos^2 x\)

Answer

\(I=\sec x\)

15) \(t\dfrac{dy}{dt}+4y=\sqrt{t} \)

Answer

\(I=t^4\)

Solve the following differential equations.

16) \(y'=3y+2\)

Answer

\(\displaystyle y=Ce^{3x}−\frac{2}{3}\). Notice that equation is also separable.

17) \(\displaystyle \frac{dy}{dx}=2y−x^2; \; y(0)=1. \)

Answer

\(\displaystyle y=\frac{1}{4} \big( e^{2x} + 2x^2+2x+1 \big) \).

18) \(xy'=3y−6x^2\)

Answer

\(y=Cx^3+6x^2\)

19) \((x+2)y'=3x+y; \; y(-1)=8; \; x>-2 \)

Answer

\(y=3(x+2)\ln (x+2) + 2x+10\)

20) \(\displaystyle \frac{dy}{dx} + y \tan x = \sec^2 x; \; y(0)=3 \)

Answer

\(\displaystyle y= \big( 3 + \sin x \big) \cos x\)

21) \(xy'=3y+x^2; \; y(1)=5.\)

Answer

\(y=5x^3-x^2\)

22) \(y'+\ln x=\dfrac{y}{x}; \, x>0.\)

Answer

\(y=x\left(C-\frac{1}{2} (\ln x)^2 \right)\)

23) \(y'+y=x,\quad y(0)=3\)

Answer

\(y=x+4e^x−1\)

24) \(xy'=y−3x^3,\quad y(1)=0\)

Answer

\(y=−\dfrac{3x}{2}(x^2−1)\)

25) \((1+x^2)y'=y−1,\quad y(0)=0\)

Answer

\(y=1−e^{\tan^{−1}x}\)

26) \((2+x)y'=y+2+x,\quad y(0)=0\)

Answer

\(y=(x+2)\ln\left(\dfrac{x+2}{2}\right)\)

27) \(\sqrt{x}y'=y+2x,\quad y(0)=1\)

Answer

\(y=2e^{2\sqrt{x}}−2x−2\sqrt{x}−1\)

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