Differential equations

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Problem: Integrate the following linear differential equation under the given initial conditions: \[ x^{\prime \prime}+x^{\prime}-2 x=e^{t}, \quad x(0)=x_{0}, x^{\prime}(0)=x_{0}{ }^{\prime} . \] clip2net.com

8.2.1.1 Differential equations

2.54 $

Problem: Solve the Cauchy problem using the operational method. \[ 2 y^{\prime \prime}+5 y^{\prime}=29 \cos t, y(0)=-1, y^{\prime}(0)=0 . \] clip2net.com

8.2.1.3 Differential equations

2.54 $

Problem: Integrate the following linear differential equation, the right side of which is presented in the graph, shown in the figure. \[ \begin{array}{l} \text { When } \quad t=0, x=0, x^{\prime}=0, \\ x^{\prime \prime}+x=f(t) . \end{array} \] clip2net.com

8.2.1.2 Differential equations

5.09 $

Problem: Solve the Cauchy problem, using the operational method. \[ 2 y^{\prime \prime}+5 y^{\prime}=29 \cos t, y(0)=-1, \quad y^{\prime}(0)=0 . \] clip2net.com

8.2.1.4 Differential equations

2.54 $

Problem: Solve the differential equation, using the operational method: \[ y^{\prime \prime}(t)+5 y(t)=e^{3 t}, \quad y(0)=y^{\prime}(0)=0 . \]

8.2.1.5 Differential equations

2.04 $

Problem: Solve the Cauchy problem using Duhamel's formula: \( y^{\prime \prime}+4 y=2 \tan x, y(0)=0, y^{\prime}(0)=0 \).

8.2.1.6 Differential equations

3.82 $

Problem: Solve the Cauchy problem, using Duhamel's formula: \( y^{\prime \prime}+4 y=2 \tan x, y(0)=1, y^{\prime}(0)=2 \).

8.2.1.7 Differential equations

3.82 $

Problem: Solve the equation: \[ \int_{0}^{t} \operatorname{ch}(t-\tau) x(\tau) d \tau=\operatorname{ch} t-\cos t \] (with respect to the function \( x(t) \) ).

8.2.1.8 Differential equations

2.04 $

Problem: Solve the Cauchy problem, using the operational method: \[ y^{\prime \prime}-5 y^{\prime}+6 y=6 t-5, \quad y(0)=1, \quad y^{\prime}(0)=3 \text {. } \]

8.2.1.9 Differential equations

0 $

Problem: Solve the Cauchy problem using Duhamel's formula: \[ y^{\prime \prime}+4 y=2 \tan x, y(0)=0, y^{\prime}(0)=0 \text {. } \]

8.2.1.10 Differential equations

3.82 $

Problem: Solve the equation \[ \int_{0}^{t} \sin (t-\tau) x(\tau) d \tau=x^{\prime \prime}+x+2 \cos t-4 \] under the given initial conditions \( x(0)=0, x^{\prime}(0)=0 \) (with respect to the function \( x(t) \) ).

8.2.1.11 Differential equations

2.54 $

Problem: Solve the differential equation, using the operational method. \[ y^{\prime \prime}+4 y=\cos t, \quad y(0)=1, \quad y^{\prime}(0)=0 . \]

8.2.1.12 Differential equations

2.04 $