With variable coefficients/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). \[ \begin{array}{l} z_{x x}^{\prime \prime}-2 \sin x \cdot z_{x y}^{\prime \prime}-\cos ^{2} x \cdot z_{y y}^{\prime \prime}-z_{x}^{\prime}+(\sin x-\cos x-1) z_{y}^{\prime}=0, \\ \left.z\right|_{x=0}=3 y,\left.\quad z_{x}^{\prime}\right|_{x=0}=5 \end{array} \]

11.5.3.1 With variable coefficients

6.47 $

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

0 $

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

3.11 $

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

6.47 $

Problem: Find the solution of the second order linar equation, satisfying the given initial conditions (the Cauchy problem). \[ \begin{array}{l} y^{2} z_{x x}+2 y \cdot z_{x y}+z_{y y}+z_{x}=0, \\ z(x ; 0)=x^{3}, \quad z_{y}(x ; 0)=-x . \end{array} \]

11.5.3.5 With variable coefficients

6.47 $

Problem: Determine the type of the differential equation, bring it to the canonical form, write down the general solution, find the solution of the Cauchy problem. \[ \begin{array}{l} y^{4} u_{x x}+2 y^{2} u_{x y}+u_{y y}-\frac{2}{y} u_{y}=0, \\ u(x, 1)=\frac{x^{3}}{3}, \quad u_{y}(x, 1)=2 x . \end{array} \]

11.5.3.6 With variable coefficients

7.77 $

Problem: Bring the following equation to the canonical form in each of the regions, where the type of the considered equation is preserved: \[ x^{2} u_{x x}+2 x y u_{x y}+y^{2} u_{y y}-2 y u_{x}+y e^{\frac{y}{x}}=0 . \]

11.5.3.7 With variable coefficients

5.7 $

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

11.5.3.8 With variable coefficients

6.47 $