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Problem: Find the solutions \( y=y(x) \) of the differential equation, that are non-identically zero in the indicated area and satisfy the given boundary conditions (the Sturm-Liouville problem). \[ \left\{\begin{array}{c} y^{\prime \prime}+\lambda y=0, \quad \pi \leq x \leq 2 \pi \\ y(\pi)=y^{\prime}(2 \pi)=0 \end{array} .\right. \]

11.1.2 Sturm-Liouville problem

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Problem: Determine the type of the nonlinear equation based on the given solution: \[ u_{x x}+u_{x x} u_{x y}+u_{y y}^{2}+u_{x}+u_{y}=1 \text {, } \] a) \( u_{1}=x^{2}-2 x y+y^{2}+2 x-3 y+1 \), b) \( u_{2}=3 x-2 y \).

11.2.1 Nonlinear equations

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Problem: Determine the type of the nonlinear equation, based on the given solution: \[ \begin{array}{l} u_{x x}^{2}+u_{x y}^{2}+u_{y y}^{2}+u_{x}+u_{y}=0, \\ u=x^{2}-2 x y+y^{2}-6 x-6 y+5 . \end{array} \]

11.2.2 Nonlinear equations

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Problem: Prove in propositional calculus (letters denote arbitrary formulas): \[ (A \rightarrow B) \rightarrow((C \vee(A \rightarrow C)) \vee B) . \]

6.1.1.6 Propositional calculus

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Problem: a) Find the general solution of the first-order partial differential equation: \[ x \frac{\partial u}{\partial x}-u \frac{\partial u}{\partial y}=0, \quad(x>0) \] b) Solve the first-order partial differential equation under the given additional conditions: \[ x \frac{\partial u}{\partial x}+y \frac{\partial u}{\partial y}=u-x^{2}-y^{2} ; y=-2, \quad u=x-x^{2} . \]

11.4.1 First order partial differential equations

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Problem: Find the general solution of the first-order equation: \[ x y \frac{\partial z}{\partial x}-y^{2} \frac{\partial z}{\partial y}+x\left(1+x^{2}\right)=0 \text {. } \]

11.4.2 First order partial differential equations

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

11.4.3 First order partial differential equations

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Problem: Solve the problem of oscillations of an infinite string using the d'Alembert method. \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=4 \frac{\partial^{2} u}{\partial x^{2}}, \\ \left.u(x, t)\right|_{t=0}=\sin \frac{x}{2},\left.\frac{\partial u}{\partial t}\right|_{t=0}=\cos \frac{x}{2},-\infty

11.5.1.1 d'Alembert method

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Problem: Solve the problem of oscillations of an infinite string using the d'Alembert method: \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=9 \frac{\partial^{2} u}{\partial x^{2}}, \\ \left.u(x, t)\right|_{t=0}=\sin 4 x,\left.\frac{\partial u}{\partial t}\right|_{t=0}=\cos 4 x,-\infty

11.5.1.2 d'Alembert method

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Problem: Find the distribution of the concentration of the diffusing substance in an infinite layer \( 0 \leq x \leq \ell \), \( -\infty

11.5.2.1 Fourier method

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Problem: The diffusing substance with concentration \( C_{0}= \) const is in an infinite layer \( -h \leq x \leq h,-\infty< \) \( y, z<+\infty \) and are kept there by impenetrable partitions, located at \( \pm h \) until the \( t=0 \) moment of time, when the partitions are removed and the diffusion process in a wider layer begins \( -H \leq x \leq \) \( H(H>h) \). Find the distribution of the concentration of the diffusing substance when \( t>0 \), if on the surfaces \( x= \pm H \) there is a mass exchange with the environment, having a constant concentration of the diffusing substance \( C_{1} \).

11.5.2.2 Fourier method

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Problem: Find the solution of the second order linear equation, satisfying the given initial conditions (the Cauchy problem). \[ \begin{array}{l} z_{x x}^{\prime \prime}-2 \sin x \cdot z_{x y}^{\prime \prime}-\cos ^{2} x \cdot z_{y y}^{\prime \prime}-z_{x}^{\prime}+(\sin x-\cos x-1) z_{y}^{\prime}=0, \\ \left.z\right|_{x=0}=3 y,\left.\quad z_{x}^{\prime}\right|_{x=0}=5 \end{array} \]

11.5.3.1 With variable coefficients

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Problem: Determine the type of the equation at the point \( (1 ; 1) \). \[ x u_{x x}+2 x y u_{x y}+y u_{y y}=\cos u^{2} . \]

11.5.3.2 With variable coefficients

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Problem: Find the solution of the second order linear equation, satisfying the given initial conditions: \[ \begin{array}{l} u_{x y}-\frac{1}{y^{2}+1} u_{x}=0, \quad y>0, \quad x>0 . \\ \left.u\right|_{x=0}=y,\left.\quad u\right|_{y=0}=x . \end{array} \]

11.5.3.3 With variable coefficients

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Problem: Find the solution of the second order linear equation, satisfying the given initial conditions (the Cauchy problem). \[ \begin{array}{l} z_{x x}-2 x z_{x y}+x^{2} z_{y y}-z_{x}+(x-1) z_{y}=0, \\ z(0, y)=y, \quad z_{x}(0, y)=y^{2} . \end{array} \]

11.5.3.4 With variable coefficients

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