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Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=2, \quad u(x, 0)=2 \cos \frac{5}{2} x, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.7 Fourier method

3.39 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=3, \quad u(0, x)=x+2, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.8 Fourier method

3.39 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}+t^{2} x^{2}, \\ a=3.5, u(x, 0)=\frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.9 Fourier method

3.91 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}+t, \quad a=1.5 \\ u(x, 0)=\frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.10 Fourier method

3.91 $

Problem: Solve the boundary-value problem for the inhomogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}+t^{2} x, \quad a=2, \\ u(0, x)=\frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.11 Fourier method

4.44 $

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=4 u_{x x}, \\ u(x, 0)=15 \sin 3 \pi x, u(0, t)=0, u_{x}(4.5, t)=0 . \end{array} \]

11.5.2.12 Fourier method

5.22 $

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=4 u_{x x}, \\ u(x, 0)=5 \sin 3 \pi x, \quad u(0, t)=u(6, t)=0 . \end{array} \]

11.5.2.13 Fourier method

4.44 $

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=8 u_{x x}, \quad u(x, 0)=5 \cos 2 \pi x+6 \cos 3 \pi x, \\ u_{x}(0, t)=u_{x}(6, t)=0, \quad a=\sqrt{8}=2 \sqrt{2}, \quad l=6 . \end{array} \]

11.5.2.14 Fourier method

5.22 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{ll} u_{t t}=25 u_{x x}, & u(x, 0)=5 \sin 3 \pi x, \\ u_{t}(x, 0)=0, & u(0, t)=u(3, t)=0 . \end{array} \]

11.5.2.15 Fourier method

3.13 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=25 u_{x x}, \quad u(0, t)=u(3, t)=0 \\ u(x, 0)=5 \sin 3 \pi x, \quad u_{t}(x, 0)=20 \pi \sin 4 \pi x . \end{array} \]

11.5.2.16 Fourier method

3.91 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{ll} u_{t t}=16 u_{x x}, & u_{x}(0, t)=u_{x}(4, t)=0, \\ u(x, 0)=0, & u_{t}(x, 0)=12 \pi \cos 3 \pi x . \end{array} \]

11.5.2.17 Fourier method

4.44 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=9 u_{x x}, \quad u(x, 0)=5 \sin 5 \pi x, \\ u_{t}(x, 0)=0, \quad u(0, t)=0, \quad u_{x}(2.5, t)=0 . \end{array} \]

11.5.2.18 Fourier method

5.22 $

Problem: Solve the problem of oscillations of the string \( 0< \) \( x

11.5.2.19 Fourier method

6.52 $

Problem: Find the temperature distribution in the rod \( 0

11.5.2.20 Fourier method

7.83 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=5 u_{x x}, \quad u(x, 0)=e^{-(x-1)^{2}}, \\ u_{t}(x, 0)=3 \cdot e^{-(x-1)^{2}}, \quad u(0, t)=u(2, t)=0 . \end{array} \]

11.5.2.21 Fourier method

4.44 $

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