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Problem: Calculate the integral of a complex variable along the closed loop: \[ \int_{\left|z-\frac{1}{2}\right|=1} \frac{i z(z-i)}{\sin \pi z} d z . \]

10.1.11 Integral of a complex variable

2.5 $

Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z|=\frac{1}{3}} \frac{3-2 z+4 z^{4}}{z^{3}} d z . \]

10.1.12 Integral of a complex variable

1.5 $

Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z|=0.05} \frac{e^{4 z}-1-\sin 4 z}{z^{3} \operatorname{sh} 16 \pi z} d z . \]

10.1.13 Integral of a complex variable

3 $

Problem: Calculate the integral of a complex variable along a closed loop: \[ \int_{|z+5|=2}\left(z \sin \frac{1}{z+5}+\frac{2 \operatorname{ch}\left(\frac{\pi i z}{4}\right)}{(z+4)^{2}(z+2)}\right) d z . \]

10.1.14 Integral of a complex variable

2.5 $

Problem: Find the Laplace transform \( F(p) \) according to the given signal: \[ f(t)=\int_{0}^{t} \tau \exp \left(-\frac{t-\tau}{2}\right) d \tau \text {. } \]

10.4.4 Laplace transform

2 $

Problem: Find the Laplace transform \( F(p) \) by the given signal \( f(t)=\theta(t-1) t e^{t} \).

10.4.5 Laplace transform

1.5 $

Problem: Find the Laplace transform \( F(p) \) by the given signal: \[ f(t)=\theta\left(t-\frac{\pi}{4}\right) \sin t \text {. } \]

10.4.6 Laplace transform

2 $

Problem: Find the signal \( f(t) \) by the given Laplace transform: \[ F(p)=\frac{1}{p-2}+\frac{e^{-p}}{p}+\frac{3 e^{-4 p}}{p^{2}+9} \]

10.4.7 Laplace transform

2.5 $

Problem: Solve the differential equation: \[ y^{\prime}=(4 x+y+1)^{2} \text {. } \]

8.1.1.3 First order differential equations

1.25 $

Problem: Write an equation, which is satisfied by all extremum points of the integral curves of the differential equation \( y^{\prime}=f(x, y) \). How to differentiate maximum points from minimum points?

8.1.1.4 First order differential equations

1.25 $

Problem: Write an equation, which is satisfied by all the inflection points of the curves of the differential equation \( y^{\prime}=f(x, y) \) and, particularly, the differential equation

8.1.1.5 First order differential equations

1 $

Problem: Solve the differential equation: \[ (2 x-y+1) d x+(2 y-x-1) d y=0 . \]

8.1.1.6 First order differential equations

2 $

Problem: Solve the differential equation: \[ y^{\prime}=\frac{y}{x+y^{3}} \text {. } \]

8.1.1.7 First order differential equations

1.25 $

Problem: Solve the differential equation: \[ y^{\prime}=\frac{2 x}{x^{2} \cos y+\sin 2 y} \text {. } \]

8.1.1.8 First order differential equations

1.5 $

Problem: Solve the differential equation: \[ y=\left(y^{\prime}\right)^{2}+4\left(y^{\prime}\right)^{3} . \]

8.1.1.9 First order differential equations

1.25 $

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