1 – oblicz i narysuj wykres funkcji z WP
2 – oblicz
3 – oblicz
1.1. $2y^{'}\sqrt{x} = 1$ y(1) = 2
$y\left( x \right) = \ c1 + \ \sqrt{x}$ $y\left( x \right) = \ \sqrt{x} + \ 1$
1.2. y’cosx – ysinx = sin2x
$$y\left( x \right) = \ c1*\frac{1}{\text{cosx}} - \frac{1}{2}\cos 2x\frac{1}{\text{cosx}}$$
1.3. y’’ + y’ – 2y = 3ex
y(x) = c1e−2x + c2ex + exx
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2.1. y’ = yctgx y(π/6) = 1
y(x) = c1 sin(x) y(x) = 2 sin(x)
2.2. $y^{'} - \ \frac{2y}{x} = \frac{e^{{- x}^{2}}}{x}$
2.3. y’’ + 9y = cosx
$$y\left( x \right) = \ c2\sin\left( 3x \right) + \ c1\cos\left( 3x \right) + \frac{\cos\left( x \right)}{8}$$
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3.1. x2y’ = 1 y(-1) = 1
$y\left( x \right) = \ c1 - \frac{1}{x}$ $\backslash ty\left( x \right) = \ - \frac{1}{x}$
3.2. xy’ + y = lnx + 1
$$y\left( x \right) = \frac{c1}{x} + \ \ln x$$
3.3. y’’ – 4y’ + 5y = x2
$$y(x)\ = \ c1\ e^{2x}\ \sin(x) + c2\ e^{2x}\ \cos(x) + \frac{x^{2}}{5} + \frac{8\ x}{25} + \frac{22}{125}$$
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4.1. y’(x2 – 4) = 2xy y(0) = 1
y(x) = c1 (x2-4) $y(x)\ = \ 1 - \frac{x^{2}}{4}$
4.2. y’ + 2y = (x + 3) e2x
$$y\left( x \right) = \ c1\ e^{- 2\ x} + \frac{1}{4}e^{2\ x}\ x + \frac{11\ e^{2\ x}}{16}$$
4.3. y’’ + 4y’ = sinx
$$y(x)\ = \ c1\ e^{- 4\ x} + c2 - \frac{\sin\left( x \right)}{17} - \frac{4\ \cos\left( x \right)}{17}$$
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5.1. y’ + ytgx = 0 y(0) = 1
y(x) = c1 cos(x) y(x) = cos(x)
5.2. $y^{'} - \ \frac{2y}{x} = x^{2} + \ 1$
y(x) = c1 x2 + x3 − x
5.3. y’’ + y’ = xex
$$y(x)\ = \ c1\ e^{- x} + c2 + \frac{e^{x}x}{2} - \frac{3\ e^{x}}{4}$$