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Problem: A particle of mass \( m \) rises vertically with an initial velocity \( v_{0} \). Assuming air resistance is proportional to the square of the velocity of the particle (proportionality coefficient \( k>0 \) ), find the height of the ascent of the particle and the velocity, with which it will return to its original position, as well as the time of ascent and descent. clip2net.com

8.1.4.15 Geometric and physical applications

3.85 $

Problem: \( \mathrm{A} \) ball with a mass of \( 400 \mathrm{~g} \) is thrown upwards with a speed of \( 20 \mathrm{~m} / \mathrm{s} \). Calculate the time of the ball's ascent and the maximum height of the ascent, if the air resistance is proportional to the square of the speed of the ball (proportionality coefficient \( k>0 \) ), moreover, it is equal to \( 0,0048 \mathrm{~N} \) when the speed is \( 1 \mathrm{~m} / \mathrm{s} \). Take \( g=10 \mathrm{~m} / \mathrm{s}^{2} \).

8.1.4.16 Geometric and physical applications

3.34 $

Problem: Find the law of linear motion of the material point \( m \) under the action of a repulsive force, inversely proportional to the cube of the distance from the point to the fixed centre (proportionality coefficient \( k>0 \) ). At the initial moment the point is at rest and at a distance \( x_{0} \) from the centre. clip2net.com

8.1.4.17 Geometric and physical applications

2.06 $

Problem: The material point \( m \) moves linearly to the fixed centre, attracting it with a force inversely proportional to the cube of the distance from the centre (proportionality coefficient \( k>0 \) ). Find the law of motion, if it starts from a state of rest, when the point is at \( x_{0} \) distance from the centre. Determine the time after which the point will reach the centre.

8.1.4.18 Geometric and physical applications

2.06 $

Problem: Determine how long it will take a particle, attracted by the Earth, to fall on the Earth, according to Newton's law (with an acceleration, inversly proportional to the square of the distance between them), if at the initial moment the velocity of the particle is equal to zero, and its distance from the centre of the Earth is equal to \( H \). Neglect the atmospheric resistance. The acceleration of the free fall on the Earth's surface is constant and equal to \( g \).

8.1.4.19 Geometric and physical applications

3.34 $

Problem: Find the integral curve of the differential equation \( y^{\prime \prime}-y=0 \) tangent at the point \( O(0,0) \) to the line \( y=x \). clip2net.com

8.1.4.20 Geometric and physical applications

1.8 $

Problem: Find the integral curve of the differential equation \( y^{\prime \prime}-4 y^{\prime}+3 y=0 \) tangent at the point \( M_{0}(0,2) \) to the line \( x-y+2=0 \). clip2net.com

8.1.4.21 Geometric and physical applications

1.8 $

Problem: Find the integral curve of the equation \( y y^{\prime \prime}+ \) \( +\left(y^{\prime}\right)^{2}-1=0 \) passing through the point \( M_{0}(0,1) \), and tangent at this point to the line \( x+y=1 \). clip2net.com

8.1.4.22 Geometric and physical applications

2.06 $

Problem: Find the surface that satisfies the first-order partial differential equation under the given conditions: \[ x \frac{\partial z}{\partial x}+z \frac{\partial z}{\partial y}=y, \quad y=2 z, \quad x+2 y=z . \]

11.4.34 First order partial differential equations

3.08 $

Problem: Find the general solution of the first-order partial differential equation: \[ \left(x+y^{2}+u^{2}\right) \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=u \text {. } \]

11.4.35 First order partial differential equations

3.85 $

Problem: Solve the Cauchy problem for the first-order partial differential equation: \[ x \frac{\partial u}{\partial x}+\left(y+x^{2}\right) \frac{\partial u}{\partial y}=u, \quad y=x, \quad u=x^{2} . \]

11.4.36 First order partial differential equations

3.85 $

Problem: Find the general integral of the first-order partial differential equation: \[ (x y-u) \frac{\partial u}{\partial x}+\left(1-y^{2}\right) \frac{\partial u}{\partial y}=y u-x . \]

11.4.37 First order partial differential equations

4.62 $

Problem: Find the solution of the first-order partial differential equation under the given conditions: \[ x y^{2} \frac{\partial u}{\partial x}+2 y^{3} \frac{\partial u}{\partial y}=2\left(y u-x^{2}\right), \quad y=1, \quad u=2 x^{2} . \]

11.4.38 First order partial differential equations

3.85 $

Problem: Find the general solution of the first-order partial differential equation: \[ 2 x \frac{\partial u}{\partial x}+(y-x) \frac{\partial u}{\partial y}-x^{2}=0 . \]

11.4.39 First order partial differential equations

3.34 $

Problem: Solve the first-order partial differential equation under the given additional conditions: \[ y \frac{\partial u}{\partial x}-x \frac{\partial u}{\partial y}=y^{2}-x^{2}, y=a, 2 u=(x+a)^{2}, a=\text { const. } \]

11.4.40 First order partial differential equations

4.62 $

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