7.8.2 2.2

7.8.2.1 [1761] Problem 1

problem number 1761

Added June 28, 2019.

Problem Chapter 8.2.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{w}_x+b{w}_y+c{w}_z=(\lambda x^2+\beta y^2+\gamma z^2+\delta )w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*D[w[x, y,z], x] + b*D[w[x, y,z], y] +c*D[w[x,y,z],z]== (alpha*x^2+beta*y^2+gamma*z^2+delta)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y-\frac{bx}{a},z-\frac{cx}{a})\exp \left(\frac{1}{3}(\frac{\alpha x^3+3\delta x}{a}+\frac{\beta y^3}{b}+\frac{\gamma z^3}{c})\right)\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a*diff(w(x,y,z),x)+b*diff(w(x,y,z),y)+c*diff(w(x,y,z),z)=  (alpha*x^2+beta*y^2+gamma*z^2+delta)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{ay-bx}{a},\frac{az-cx}{a}){\mathrm{e}}^{\frac{((\frac{\alpha x^2}{3}+\beta y^2+\gamma z^2+\delta )a^2-(b\beta y+c\gamma z)ax+\frac{(b^2\beta +c^2\gamma )x^2}{3})x}{a^3}}$$

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7.8.2.2 [1762] Problem 2

problem number 1762

Added June 28, 2019.

Problem Chapter 8.2.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+(a_1{x}^2+a_0)w_y+(b_1{x}^2+b_0)w_z=(\lambda x+\beta y+\gamma z+\delta )w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x^2+a0)*D[w[x, y,z], y] +(b1*x^2+b0)*D[w[x,y,z],z]== (alpha*x+beta*y+gamma*z+delta)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(-\text{a0}x-\frac{\text{a1}{x}^3}{3}+y,-\text{b0}x-\frac{\text{b1}{x}^3}{3}+z)\exp (-\frac{1}{4}x(2\text{a0}\beta x+\text{a1}\beta x^3-2\alpha x+2\text{b0}\gamma x+\text{b1}\gamma x^3-4\beta y-4\delta -4\gamma z))\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   diff(w(x,y,z),x)+(a1*x^2+a0)*diff(w(x,y,z),y)+(b1*x^2+b0)*diff(w(x,y,z),z)=  (alpha*x+beta*y+gamma*z+delta)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(-\frac{1}{3}\mathit{a}\mathit{1}{x}^3-\mathit{a}\mathit{0}x+y,-\frac{1}{3}\mathit{b}\mathit{1}{x}^3-\mathit{b}\mathit{0}x+z){\mathrm{e}}^{-\frac{(-2\alpha x+(\mathit{a}\mathit{1}{x}^3+2\mathit{a}\mathit{0}x-4y)\beta -4\delta +(\mathit{b}\mathit{1}{x}^3+2\mathit{b}\mathit{0}x-4z)\gamma )x}{4}}$$

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7.8.2.3 [1763] Problem 3

problem number 1763

Added June 28, 2019.

Problem Chapter 8.2.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+(ay+k_1{x}^2+k_0)w_y+(bz+s_1{x}^2+s_0)w_z=(c_1{x}^2+c_0)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a*y+k1*x^2+k0)*D[w[x, y,z], y] +(b*z+s1*x^2+s0)*D[w[x,y,z],z]== (c1*x^2+c0)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{\text{c0}x+\frac{\text{c1}{x}^3}{3}}{c}_1(\frac{e^{-ax}(a^2(\text{k0}+\text{k1}{x}^2)+a^{3}y+2a\text{k1}x+2\text{k1})}{a^3},\frac{e^{-bx}(b^2(\text{s0}+\text{s1}{x}^2)+b^{3}z+2b\text{s1}x+2\text{s1})}{b^3})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   diff(w(x,y,z),x)+(a*y+k1*x^2+k0)*diff(w(x,y,z),y)+(b*z+s1*x^2+s0)*diff(w(x,y,z),z)=  (c1*x^2+c0)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{(a^{3}y+2a\mathit{k}\mathit{1}x+(\mathit{k}\mathit{1}{x}^2+\mathit{k}\mathit{0})a^2+2\mathit{k}\mathit{1}){\mathrm{e}}^{-ax}}{a^3},\frac{(b^{3}z+2b\mathit{s}\mathit{1}x+(\mathit{s}\mathit{1}{x}^2+\mathit{s}\mathit{0})b^2+2\mathit{s}\mathit{1}){\mathrm{e}}^{-bx}}{b^3}){\mathrm{e}}^{\frac{(\mathit{c}\mathit{1}{x}^2+3\mathit{c}\mathit{0})x}{3}}$$

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7.8.2.4 [1764] Problem 4

problem number 1764

Added June 28, 2019.

Problem Chapter 8.2.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$w_x+(a_{2}xy+a_1{x}^2+a_0)w_y+(b_{2}xy+b_1{x}^2+b_0)w_z=(c_{2}y+c_{1}z+c_{0}x+s)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y,z], x] + (a1*x*y+a1*x^2+a0)*D[w[x, y,z], y] +(b2*x*y+b1*x^2+b0)*D[w[x,y,z],z]== (c2*x+c1*z+c0*x+s)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(\frac{\text{a0}\text{b2}x}{\text{a1}}-\frac{\text{b2}y}{\text{a1}}-\text{b0}x-\frac{\text{b1}{x}^3}{3}+\frac{\text{b2}{x}^3}{3}+z,e^{-\frac{\text{a1}{x}^2}{2}}(x+y)-\frac{\sqrt{\frac{\pi }{2}}(\text{a0}+1)\text{erf}\left(\frac{\sqrt{\text{a1}}x}{\sqrt{2}}\right)}{\sqrt{\text{a1}}})\exp \left(\frac{2(\text{a0}+1)\text{a1}\text{b2}\text{c1}{x}^2\mathrm{HypergeometricPFQ}(\{1,1\},\{\frac{3}{2},2\},\frac{\text{a1}{x}^2}{2})+2\text{b2}\text{c1}{e}^{-\frac{\text{a1}{x}^2}{2}}\mathrm{Erfi}\left(\frac{\sqrt{\text{a1}}x}{\sqrt{2}}\right)(\sqrt{2\pi }\sqrt{\text{a1}}(x+y)-\pi (\text{a0}+1)e^{\frac{\text{a1}{x}^2}{2}}\text{erf}\left(\frac{\sqrt{\text{a1}}x}{\sqrt{2}}\right))+\text{a1}x(2\text{b2}\text{c1}((\text{a0}-1)x-2y)+\text{a1}(-2\text{b0}\text{c1}x-\text{b1}\text{c1}{x}^3+\text{b2}\text{c1}{x}^3+2\text{c0}x+4\text{c1}z+2\text{c2}x+4s))}{4{\text{a1}}^2}\right)\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   diff(w(x,y,z),x)+(a1*x*y+a1*x^2+a0)*diff(w(x,y,z),y)+(b2*x*y+b1*x^2+b0)*diff(w(x,y,z),z)=  (c2*x+c1*z+c0*x+s)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{(x+y)\sqrt{\mathit{a}\mathit{1}}{\mathrm{e}}^{-\frac{\mathit{a}\mathit{1}{x}^2}{2}}-\frac{\sqrt{\pi }\sqrt{2}(\mathit{a}\mathit{0}+1)\mathrm{erf}\left(\frac{\sqrt{2}\sqrt{\mathit{a}\mathit{1}}x}{2}\right)}{2}}{\sqrt{\mathit{a}\mathit{1}}},-\frac{3(-\frac{\sqrt{2}(\mathit{a}\mathit{0}+1)(\frac{\sqrt{\pi }\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}}{\sqrt{\mathit{a}\mathit{1}}}-1)\mathrm{erf}\left(\frac{\sqrt{2}\sqrt{\mathit{a}\mathit{1}}x}{2}\right)}{2}+\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}(x+y){\mathrm{e}}^{-\frac{\mathit{a}\mathit{1}{x}^2}{2}})\mathit{b}\mathit{2}{\mathrm{e}}^{\frac{\mathit{a}\mathit{1}{x}^2}{2}}+\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}(-(\mathit{a}\mathit{1}{x}^2+3\mathit{a}\mathit{0}+3)\mathit{b}\mathit{2}x+(\mathit{b}\mathit{1}{x}^3+3\mathit{b}\mathit{0}x-3z)\mathit{a}\mathit{1})}{3\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}\mathit{a}\mathit{1}}){\mathrm{e}}^{{\int }_{}^x\frac{-2({\mathit{\text{\_a}}}^2\mathit{a}\mathit{1}+3\mathit{a}\mathit{0}+3)\mathit{\text{\_a}}\mathit{b}\mathit{2}\mathit{c}\mathit{1}-3((-2x-2y){\mathrm{e}}^{-\frac{\mathit{a}\mathit{1}{x}^2}{2}}+\frac{\sqrt{2}\sqrt{\pi }(\mathit{a}\mathit{0}+1)\mathrm{erf}\left(\frac{\sqrt{2}\sqrt{\mathit{a}\mathit{1}}x}{2}\right)}{\sqrt{\mathit{a}\mathit{1}}})\mathit{b}\mathit{2}\mathit{c}\mathit{1}{\mathrm{e}}^{\frac{{\mathit{\text{\_a}}}^2\mathit{a}\mathit{1}}{2}}+6(\frac{{\mathit{\text{\_a}}}^3\mathit{b}\mathit{1}\mathit{c}\mathit{1}}{3}+(\mathit{b}\mathit{0}\mathit{c}\mathit{1}+\mathit{c}\mathit{0}+\mathit{c}\mathit{2})\mathit{\text{\_a}}+s)\mathit{a}\mathit{1}-\frac{2(-\frac{3\sqrt{2}(\mathit{a}\mathit{0}+1)\mathit{b}\mathit{2}\mathrm{erf}\left(\frac{\sqrt{2}\mathit{\text{\_a}}\sqrt{\mathit{a}\mathit{1}}}{2}\right){\mathrm{e}}^{\frac{{\mathit{\text{\_a}}}^2\mathit{a}\mathit{1}}{2}}}{2}-\frac{3\sqrt{2}(\mathit{a}\mathit{0}+1)(\frac{\sqrt{\pi }\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}}{\sqrt{\mathit{a}\mathit{1}}}-1)\mathit{b}\mathit{2}\mathrm{erf}\left(\frac{\sqrt{2}\sqrt{\mathit{a}\mathit{1}}x}{2}\right){\mathrm{e}}^{\frac{\mathit{a}\mathit{1}{x}^2}{2}}}{2}+3\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}(x+y)\mathit{b}\mathit{2}{\mathrm{e}}^{-\frac{\mathit{a}\mathit{1}{x}^2}{2}}{\mathrm{e}}^{\frac{\mathit{a}\mathit{1}{x}^2}{2}}+\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}(-(\mathit{a}\mathit{1}{x}^2+3\mathit{a}\mathit{0}+3)\mathit{b}\mathit{2}x+(\mathit{b}\mathit{1}{x}^3+3\mathit{b}\mathit{0}x-3z)\mathit{a}\mathit{1}))\mathit{c}\mathit{1}}{\sqrt{\frac{\mathit{a}\mathit{1}}{\pi }}}}{6\mathit{a}\mathit{1}}d\mathit{\text{\_a}}}$$

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7.8.2.5 [1765] Problem 5

problem number 1765

Added June 28, 2019.

Problem Chapter 8.2.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$ax{w}_x+by{w}_y+cz{w}_z=x(\lambda x+\beta y+\gamma z)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x*D[w[x, y,z], x] + b*y*D[w[x, y,z], y] +c*z*D[w[x,y,z],z]== x*(lambda*x+beta*y+gama*z)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}})e^{\frac{\beta xy}{a+b}+\frac{\text{gama}xz}{a+c}+\frac{\lambda x^2}{2a}}\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a*x*diff(w(x,y,z),x)+b*y*diff(w(x,y,z),y)+c*z*diff(w(x,y,z),z)=  x*(lambda*x+beta*y+gama*z)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}}){\mathrm{e}}^{\frac{(bc\lambda x+(2\beta y+2\mathit{gama}z+\lambda x)a^2+(2b\mathit{gama}z+2\beta cy+(b+c)\lambda x)a)x}{2(a+b)(a+c)a}}$$

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7.8.2.6 [1766] Problem 6

problem number 1766

Added June 28, 2019.

Problem Chapter 8.2.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{x}^2{w}_x+bxy{w}_y+cxz{w}_z=(\lambda x+\beta y+\gamma z)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y,z], x] + b*x*y*D[w[x, y,z], y] +c*x*z*D[w[x,y,z],z]== (lambda*x+beta*y+gama*z)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to x^{\frac{\lambda }{a}}{e}^{-\frac{\frac{\beta y}{a-b}+\frac{\text{gama}z}{a-c}}{x}}{c}_1(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a*x^2*diff(w(x,y,z),x)+b*x*y*diff(w(x,y,z),y)+c*x*z*diff(w(x,y,z),z)=  (lambda*x+beta*y+gama*z)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=x^{\frac{\lambda }{a}}\mathit{\text{\_F1}}(y{x}^{-\frac{b}{a}},z{x}^{-\frac{c}{a}}){\mathrm{e}}^{\frac{-(a-c)\beta y-(a-b)\mathit{gama}z}{(a-b)(a-c)x}}$$

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7.8.2.7 [1767] Problem 7

problem number 1767

Added June 28, 2019.

Problem Chapter 8.2.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{x}^2{w}_x+bxy{w}_y+c{z}^2{w}_z=k{y}^{2}w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y,z], x] + b*x*y*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*y^2*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to e^{-\frac{k{y}^2}{ax-2bx}}{c}_1(y{x}^{-\frac{b}{a}},\frac{c}{ax}-\frac{1}{z})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a*x^2*diff(w(x,y,z),x)+b*x*y*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)=  k*y^2*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(y{x}^{-\frac{b}{a}},\frac{ax-cz}{axz}){\mathrm{e}}^{-\frac{k{y}^2}{(a-2b)x}}$$

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7.8.2.8 [1768] Problem 8

problem number 1768

Added June 28, 2019.

Problem Chapter 8.2.2.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$

$$a{x}^2{w}_x+b{y}^2{w}_y+c{z}^2{w}_z=kxyw$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y,z], x] + b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== k*x*y*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to {\left(\frac{ax}{y}\right)}^{\frac{kxy}{ax-by}}{c}_1(\frac{b}{ax}-\frac{1}{y},\frac{c}{ax}-\frac{1}{z})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a*x^2*diff(w(x,y,z),x)+b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)=  k*x*y*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)={\left(\frac{ax}{y}\right)}^{\frac{kxy}{ax-by}}\mathit{\text{\_F1}}(\frac{ax-by}{axy},\frac{ax-cz}{axz})$$

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7.8.2.9 [1769] Problem 9

problem number 1769

Added June 28, 2019.

Problem Chapter 8.2.2.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y,z)$ $$a{x}^2{w}_x+b{y}^2{w}_y+c{z}^2{w}_z=(\lambda x^2+\beta y^2+\gamma z^2)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^2*D[w[x, y,z], x] + b*y^2*D[w[x, y,z], y] +c*z^2*D[w[x,y,z],z]== (lambda*x^2+beta*y^2+gamma*z^2)*w[x,y,z]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
 

$$\left\{\{w(x,y,z)\to c_1(\frac{b}{ax}-\frac{1}{y},\frac{c}{ax}-\frac{1}{z})\exp (\frac{\beta y^2}{by-ax}+\frac{\gamma z^2}{cz-ax}+\frac{\lambda x}{a})\}\right\}$$

Maple ✓

restart; 
local gamma; 
pde :=   a*x^2*diff(w(x,y,z),x)+b*y^2*diff(w(x,y,z),y)+c*z^2*diff(w(x,y,z),z)=  (lambda*x^2+beta*y^2+gamma*z^2)*w(x,y,z); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
 

$$w(x,y,z)=\mathit{\text{\_F1}}(\frac{ax-by}{axy},\frac{ax-cz}{axz}){\mathrm{e}}^{\frac{bc\lambda xyz+(-\beta y^2-\gamma z^2+\lambda x^2)a^{2}x+(\beta c{y}^{2}z-c\lambda x^{2}z-(-\gamma z^2+\lambda x^2)by)a}{(ax-cz)(ax-by)a}}$$

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