7.4.7 3.2

7.4.7.1 [1071] Problem 1

problem number 1071

Added Feb. 23, 2019.

Problem Chapter 4.3.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{w}_x+b{w}_y=(cy{e}^{\lambda x}+kx{e}^{\mu y})w$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1(y-\frac{bx}{a})\exp (-\frac{bc{e}^{\lambda x}}{a^2{\lambda }^2}-\frac{ak{e}^{\mu y}}{b^2{\mu }^2}+\frac{cy{e}^{\lambda x}}{a\lambda }+\frac{kx{e}^{\mu y}}{b\mu })\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right){\mathrm{e}}^{\frac{-(-b\mu x+a)a^{2}k{\lambda }^2{\mathrm{e}}^{\mu y}+(a\lambda y-b)b^{2}c{\mu }^2{\mathrm{e}}^{\lambda x}}{a^2{b}^2{\lambda }^2{\mu }^2}}$$

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7.4.7.2 [1072] Problem 2

problem number 1072

Added Feb. 23, 2019.

Problem Chapter 4.3.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$x{w}_x+y{w}_y=ax{e}^{\lambda x+\mu y}w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*Exp[lambda*x + mu*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1\left(\frac{y}{x}\right)e^{\frac{ax{e}^{\lambda x+\mu y}}{\lambda x+\mu y}}\}\right\}$$

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+y*diff(w(x,y),y) =   a*x*exp(lambda*x+mu*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{y}{x}\right){\mathrm{e}}^{\frac{a{\mathrm{e}}^{\lambda x+\mu y}}{\lambda +\frac{\mu y}{x}}}$$

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7.4.7.3 [1073] Problem 3

problem number 1073

Added Feb. 23, 2019.

Problem Chapter 4.3.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$x{w}_x+y{w}_y=(ay{e}^{\lambda x}+bx{e}^{\mu y})w$$

Mathematica ✓

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

$$\left\{\{w(x,y)\to c_1\left(\frac{y}{x}\right)e^{\frac{ay{e}^{\lambda x}}{\lambda x}+\frac{bx{e}^{\mu y}}{\mu y}}\}\right\}$$

Maple ✓

restart; 
pde := x*diff(w(x,y),x)+y*diff(w(x,y),y) =    (a*y*exp(lambda*x)+ b*x*exp(mu*y))*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{y}{x}\right){\mathrm{e}}^{\frac{(\frac{a\mu y^2{\mathrm{e}}^{\lambda x}}{x^2}+b\lambda {\mathrm{e}}^{\mu y})x}{\lambda \mu y}}$$

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7.4.7.4 [1074] Problem 4

problem number 1074

Added Feb. 23, 2019.

Problem Chapter 4.3.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{x}^k{w}_x+b{e}^{\lambda y}{w}_y=(c{x}^n+s)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*x^k*D[w[x, y], x] + b*Exp[lambda*y]*D[w[x, y], y] == (c*x^n + s)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to e^{\frac{x^{1-k}(\frac{c{x}^n}{-k+n+1}+\frac{s}{1-k})}{a}}{c}_1(\frac{b{x}^{1-k}}{a(k-1)}-\frac{e^{-\lambda y}}{\lambda })\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{b\lambda x^{-k+1}-(k-1)a{\mathrm{e}}^{-\lambda y}}{(k-1)b\lambda }\right){\mathrm{e}}^{-\frac{((k-1)c{x}^n+(k-n-1)s)x^{-k+1}}{(k-1)(k-n-1)a}}$$

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7.4.7.5 [1075] Problem 5

problem number 1075

Added Feb. 23, 2019.

Problem Chapter 4.3.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{y}^k{w}_x+b{e}^{\lambda x}{w}_y=(c{e}^{\mu x}+s)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*y^k*D[w[x, y], x] + b*Exp[lambda*x]*D[w[x, y], y] == (c*Exp[mu*x] + s)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1(\frac{y^{k+1}}{k+1}-\frac{b{e}^{\lambda x}}{a\lambda })\exp (-\frac{y^{k+1}{\left({\left(y^{k+1}\right)}^{\frac{1}{k+1}}\right)}^{-k}(c\lambda e^{\mu x}{}_2{F}_1(1,\frac{\lambda +k\mu +\mu }{k\lambda +\lambda };\frac{\lambda +\mu }{\lambda };\frac{b{e}^{\lambda x}(k+1)}{b{e}^{\lambda x}(k+1)-a\lambda y^{k+1}})-(k+1)\mu s{}_2{F}_1(1,\frac{1}{k+1};\frac{k+2}{k+1};\frac{a\lambda y^{k+1}}{a\lambda y^{k+1}-b{e}^{\lambda x}(k+1)}))}{\mu (b(k+1)e^{\lambda x}-a\lambda y^{k+1})})\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{a\lambda y{y}^k-(k+1)b{\mathrm{e}}^{\lambda x}}{a\lambda }\right){\mathrm{e}}^{{\int }_{}^x\frac{(c{\mathrm{e}}^{\mathit{\text{\_a}}\mu }+s){\left({\left(\frac{a\lambda y^{k+1}+(k+1)b{\mathrm{e}}^{\mathit{\text{\_a}}\lambda }-(k+1)b{\mathrm{e}}^{\lambda x}}{a\lambda }\right)}^{\frac{1}{k+1}}\right)}^{-k}}{a}d\mathit{\text{\_a}}}$$

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7.4.7.6 [1076] Problem 6

problem number 1076

Added Feb. 23, 2019.

Problem Chapter 4.3.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{e}^{\lambda x}{w}_x+b{y}^k{w}_y=(c{x}^n+s)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[lambda*x]*D[w[x, y], x] + b*y^k*D[w[x, y], y] == (c*x^n + s)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1(\frac{b{e}^{-\lambda x}}{a\lambda }-\frac{y^{1-k}}{k-1})\exp (-\frac{c{x}^n(\lambda x{)}^{-n}\mathrm{Gamma}(n+1,\lambda x)+s{e}^{-\lambda x}}{a\lambda })\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{a\lambda y^{-k+1}-(k-1)b{\mathrm{e}}^{-\lambda x}}{a\lambda }\right){\mathrm{e}}^{\frac{c{x}^n{\left(\lambda x\right)}^{-\frac{n}{2}}\mathrm{WhittakerM}(\frac{n}{2},\frac{n}{2}+\frac{1}{2},\lambda x){\mathrm{e}}^{-\frac{\lambda x}{2}}-({\mathrm{e}}^{-\lambda x}-1)(n+1)s}{(n+1)a\lambda }}$$

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7.4.7.7 [1077] Problem 7

problem number 1077

Added Feb. 23, 2019.

Problem Chapter 4.3.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for $w(x,y)$ $$a{e}^{\lambda y}{w}_x+b{x}^k{w}_y=(c{e}^{\mu x}+s)w$$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  a*Exp[lambda*y]*D[w[x, y], x] + b*x^k*D[w[x, y], y] == (c*Exp[mu*x] + s)*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$$\left\{\{w(x,y)\to c_1(\frac{e^{\lambda y}}{\lambda }-\frac{b{x}^{k+1}}{ak+a})\exp ({\int }_1^x\frac{(k+1)(e^{\mu K[1]}c+s)}{a{e}^{\lambda y}(k+1)+b\lambda (K[1{]}^{k+1}-x^{k+1})}dK[1])\}\right\}$$

Maple ✓

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

$$w(x,y)=\mathit{\text{\_F1}}\left(\frac{-b\lambda x^{k+1}+(k+1)a{\mathrm{e}}^{\lambda y}}{(k+1)b\lambda }\right){\mathrm{e}}^{{\int }_{}^x\frac{(c{\mathrm{e}}^{\mathit{\text{\_a}}\mu }+s)(k+1)}{b\lambda {\mathit{\text{\_a}}}^{k+1}-b\lambda x^{k+1}+(k+1)a{\mathrm{e}}^{\lambda y}}d\mathit{\text{\_a}}}$$

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