6.5.25 8.3

6.5.25.1 [1361] Problem 1
6.5.25.2 [1362] Problem 2
6.5.25.3 [1363] Problem 3
6.5.25.4 [1364] Problem 4
6.5.25.5 [1365] Problem 5
6.5.25.6 [1366] Problem 6

6.5.25.1 [1361] Problem 1

problem number 1361

Added April 13, 2019.

Problem Chapter 5.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ x w_x + y w_y = x f(\frac {y}{x}) w + g(x,y) \]

Mathematica

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + y*D[w[x, y], y] == x*f[y/x]*w[x,y]+g[x,y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to e^{x f\left (\frac {y}{x}\right )} \left (\int _1^x\frac {e^{-f\left (\frac {y}{x}\right ) K[1]} g\left (K[1],\frac {y K[1]}{x}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]

Maple

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\mathit {\_a} f \left (\frac {y}{x}\right )} g \left (\mathit {\_a} , \frac {\mathit {\_a} y}{x}\right )}{\mathit {\_a}}d\mathit {\_a} +\mathit {\_F1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{x f \left (\frac {y}{x}\right )}\]

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6.5.25.2 [1362] Problem 2

problem number 1362

Added April 13, 2019.

Problem Chapter 5.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ a x w_x + b y w_y = f(x,y) w + g(x,y) \]

Mathematica

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

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right ) g\left (K[2],x^{-\frac {b}{a}} y K[2]^{\frac {b}{a}}\right )}{a K[2]}dK[2]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]

Maple

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

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {{\mathrm e}^{-\frac {\int \frac {f \left (\mathit {\_b} , y \mathit {\_b}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_b}}d \mathit {\_b}}{a}} g \left (\mathit {\_b} , y \mathit {\_b}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_b} a}d\mathit {\_b} +\mathit {\_F1} \left (y x^{-\frac {b}{a}}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {f \left (\mathit {\_a} , y \mathit {\_a}^{\frac {b}{a}} x^{-\frac {b}{a}}\right )}{\mathit {\_a} a}d\mathit {\_a}}\]

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6.5.25.3 [1363] Problem 3

problem number 1363

Added April 13, 2019.

Problem Chapter 5.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ f(x) w_x + g(x) w_y = h(x,y) w + F(x,y) \]

Mathematica

ClearAll["Global`*"]; 
pde = f[x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h\left (K[2],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {h\left (K[2],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) F\left (K[3],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[3]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[3])}dK[3]+c_1\left (y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right )\right )\right \}\right \}\]

Maple

restart; 
pde :=  f(x)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {F \left (\mathit {\_f} , y +\int \frac {g \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{-\left (\int \frac {h \left (\mathit {\_f} , y +\int \frac {g \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} \right )}}{f \left (\mathit {\_f} \right )}d\mathit {\_f} +\mathit {\_F1} \left (y -\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\mathit {\_b} , y +\int \frac {g \left (\mathit {\_b} \right )}{f \left (\mathit {\_b} \right )}d \mathit {\_b} -\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )\right )}{f \left (\mathit {\_b} \right )}d\mathit {\_b}}\]

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6.5.25.4 [1364] Problem 4

problem number 1364

Added April 13, 2019.

Problem Chapter 5.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ f(x) w_x + (g_1(x) y + g_0(x)) w_y = h(x,y) w + F(x,y) \]

Mathematica

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h\left (K[3],\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {h\left (K[3],\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[3])}dK[3]\right ) F\left (K[4],\exp \left (\int _1^{K[4]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[4]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[4])}dK[4]+c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )\right \}\right \}\]

Maple

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {F \left (\mathit {\_g} , \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} -\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g}}\right ) {\mathrm e}^{-\left (\int \frac {h \left (\mathit {\_g} , \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} -\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g}}\right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )}}{f \left (\mathit {\_g} \right )}d\mathit {\_g} +\mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}-\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\mathit {\_f} , \left (y \,{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} \right )} \mathit {g0} \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {{\mathrm e}^{-\left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right ) {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f}}\right )}{f \left (\mathit {\_f} \right )}d\mathit {\_f}}\]

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6.5.25.5 [1365] Problem 5

problem number 1365

Added April 13, 2019.

Problem Chapter 5.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ f(x) w_x + (g_1(x) y + g_0(x) y^k) w_y = h(x,y) w + F(x,y) \]

Mathematica

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*y^k)D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

$Aborted

Maple

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*y^k)*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {F \left (\mathit {\_g} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (-k +1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g}}\right ) {\mathrm e}^{-\left (\int \frac {h \left (\mathit {\_g} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (-k +1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )} \mathit {g0} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_g} \right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g}}\right )}{f \left (\mathit {\_g} \right )}d \mathit {\_g} \right )}}{f \left (\mathit {\_g} \right )}d\mathit {\_g} +\mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\mathit {\_f} , \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )}+\left (-k +1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} \right )} \mathit {g0} \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f} \right )+\left (k -1\right ) \left (\int \frac {{\mathrm e}^{\left (k -1\right ) \left (\int \frac {\mathit {g1} \left (x \right )}{f \left (x \right )}d x \right )} \mathit {g0} \left (x \right )}{f \left (x \right )}d x \right )\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\mathit {g1} \left (\mathit {\_f} \right )}{f \left (\mathit {\_f} \right )}d \mathit {\_f}}\right )}{f \left (\mathit {\_f} \right )}d\mathit {\_f}}\]

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6.5.25.6 [1366] Problem 6

problem number 1366

Added April 13, 2019.

Problem Chapter 5.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\)

\[ f(x) w_x + (g_1(x) y + g_0(x) e^{\lambda y}) w_y = h(x,y) w + F(x,y) \]

Mathematica

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*Exp[lambda*y])D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde :=  f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*exp(lambda*y))*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()