☰ matematika2

Zadaci za vježbu DIFERENCIJALNE JEDNADŽBE METODA NAJMANJIH KVADRATA I

Rješenja zadataka za vježbu

1.

Ne.

2.

$\displaystyle -xyy^{\prime}+y^2=4$ .

3.

$\displaystyle y=-\cos x$ .

4.

a): $\displaystyle 0<P<4200$ ,
b): $\displaystyle P>4200$ .

5.

$\displaystyle t=88.93$ .

6.

a): $\displaystyle m(t)=18\cdot e^{-\frac{1}{25}t\ln 2}$ ,
b): $\displaystyle t=25\frac{\ln 9}{\ln 2}$ .

7.

$\displaystyle y=ce^{-\frac{1}{x^2}}$ .

8.

$\displaystyle y=\frac{1}{1-cx}$ .

9.

$\displaystyle y=\ln \left[c(1+x^2)-1\right]$ ,

, $y=\ln (2x^2+1)$ .

10.

a): $\displaystyle 2y^2\ln (cy)+x^2=0$ ,
b): $\displaystyle \frac{y}{x}\mathop{\mathrm{arctg}}\nolimits \frac{y}{x}=\frac{1}{2}\ln (x^2+y^2)+\ln c$ ,
c): $\displaystyle (x+y-1)^3=c(x-y+3)$ ,
d): $\displaystyle 3x+y+2\ln \vert x+y-1\vert=c$ ,
e): $\displaystyle \frac{1}{3}x^3-xy+\frac{1}{3}y^3=c$ ,
f): $\displaystyle x^2\ln y+\frac{1}{3}(y^2+1)^{\frac{3}{2}}=c$ ,
g): $\displaystyle y=\left( x \sin y+y \cos y-\sin y\right)e^x=c$ .

11.

$\displaystyle x^2+(y-1)^2=1$ .

12.

$\displaystyle 2x^2+y^2=c^2$ .

13.

$\displaystyle x^2-y^2=c$ .

14.

$\displaystyle y=1$ ,

.

15.

$\displaystyle y=\sin x-1+ce^{-\sin x}$ ,

, $y=\sin x-1+e^{1-\sin x}$ .

16.

$\displaystyle y=2+c\left( 1-x^{2}\right)$ ,

, $y=2-3\left( 1-x^{2}\right)$ .

17.

a): $\displaystyle xy=\left( x^{3}+c\right) e^{-x}$ ,
b): $\displaystyle y=cx+x\sin x$ ,
c): $\displaystyle y=\frac{1}{\sqrt{1+x^{2}+ce^{x^{2}}}}$ ,
d): $\displaystyle c=\frac{1}{\sqrt[3]{c\cos ^{3}x-3\sin x\cos ^{2}x}}$ ,
e): $\displaystyle y=\frac{1}{x\sqrt[3]{3\ln \frac{c}{x}}}$ .

18.

$\displaystyle y=\frac{1}{2\left( \ln x+1\right) +cx}$ ,

, $\displaystyle y=\frac{1}{2(\ln x+1)-x}$ .

19.

$\displaystyle y\approx 1.05484$ .

20.

$\displaystyle y\approx 4.28982$ .

21.

a): $\displaystyle y=\frac{1}{3}\sin ^3 x+c_1x+c_2$ .
b): $\displaystyle y=c_1\ln (x+1)+c_2$ .
c): $\displaystyle 3x=y^3+c_1y+c_2$ .
d): $\displaystyle y=c_1e^x+c_2e^{-\frac{1}{2}x}$ .
e): $\displaystyle y=e^{-3x}(c_1\cos 2x+c_2\sin 2x)$ .

22.

$\displaystyle y=e^{-x}(c_1\cos 2x+c_2\sin 2x)$ , $y=\frac{1}{2}e^{-x}\sin 2x$ .

23.

a): $\displaystyle y=c_1+c_2e^{2x}-\frac{x^3}{6}$ .
b): $\displaystyle y=c_1e^x+c_2xe^x+e^{2x}$ .
c): $\displaystyle y=(c_1\cos 3x+c_2\sin 3x)e^x+\cos 3x-6\sin 3x$ .
d): $\displaystyle y=c_1e^x+c_2e^{-\frac{1}{2}x}+\left(\frac{4}{5}x-\frac{28}{25}\right)e^{2x}$ .

24.

$\displaystyle y(x)=c_1e^{3x}+\left(c_2-\frac{x}{4}\right)e^{-3x}+c_3\cos x+c_4\sin x$ .

25.

$\displaystyle y(x)=c_1e^{2x}+c_2xe^{2x}+\frac{1}{8}\cos 2x+\frac{1}{2}x^2e^{2x}$ .

26.

$\displaystyle y(x)=\frac{4}{15}x^{\frac{5}{2}}e^{-x}+Be^{-x}+Axe^{-x}$ .

27.

$\displaystyle y(x)=\frac{\cos ^2 x}{\sin x}-\frac{1}{2\sin x}+A\sin x+B\cos x$ .

28.

$\displaystyle y(x)=e^{-2x}\left(\frac{1}{2}x^2\ln x-\frac{3}{4}x^2+Ax+B\right)$ .

29.

$\displaystyle y(x)=Ae^{2x}+Be^x+e^x(2x^2+x)$ .

30.

$\displaystyle y(x)=c_1+c_2e^{2x}-\frac{1}{4}(x^2+x)$ , $\displaystyle z(x)=c_2e^{2x}-c_1+\frac{1}{4}(x^2-x-1)$ .

31.

$\displaystyle y(t)=-e^{-t}+\frac{1}{4}e^{-2t}+\frac{3}{4}e^{2t}$ , $\displaystyle z(t)=-\frac{2}{3}e^t-\frac{1}{12}e^{-2t}+\frac{3}{4}e^{2t}$ .

32.

$\displaystyle y(x)=c_1+c_2x+2\sin x$ , $\displaystyle z(t)=-2c_1-c_2(2x+1)-3\sin x-2\cos x$ .

33.

a)

,

, b) $\displaystyle \frac{dL}{dA}=\frac{-0.5L+0.0001AL}{2A-0.01AL}$ .

Zadaci za vježbu DIFERENCIJALNE JEDNADŽBE METODA NAJMANJIH KVADRATA I