6.3.28 8.4

6.3.28.1 [1013] Problem 1

problem number 1013

Added Feb. 17, 2019.

Problem Chapter 3.8.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

Solve for \(w(x,y)\) \[ w_x + a w_y = f(x,y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^xf(K[1],-a x+y+a K[1])dK[1]+c_1(y-a x)\right \}\right \}\]

Maple ✓

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

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

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6.3.28.2 [1014] Problem 2

problem number 1014

Added Feb. 17, 2019.

Problem Chapter 3.8.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.28.3 [1015] Problem 3

problem number 1015

Added Feb. 17, 2019.

Problem Chapter 3.8.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.28.4 [1016] Problem 4

problem number 1016

Added Feb. 17, 2019.

Problem Chapter 3.8.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.3.28.5 [1017] Problem 5

problem number 1017

Added Feb. 17, 2019.

Problem Chapter 3.8.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

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

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

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6.3.28.6 [1018] Problem 6

problem number 1018

Added Feb. 17, 2019.

Problem Chapter 3.8.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

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

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6.3.28.7 [1019] Problem 7

problem number 1019

Added Feb. 17, 2019.

Problem Chapter 3.8.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y = h(x,y) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  f1[x]*g1[y]*D[w[x, y], x] + f2[x]*g2[y]*D[w[x, y], y] == h[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  f1(x)*g1(y)*diff(w(x,y),x) +f2(x)*g2(y)*diff(w(x,y),y) =  h(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {h \left (\mathit {\_f} , \RootOf \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {g1} \left (\mathit {\_a} \right )}{\mathit {g2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{\mathit {f1} \left (\mathit {\_f} \right ) \mathit {g1} \left (\RootOf \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {g1} \left (\mathit {\_a} \right )}{\mathit {g2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d\mathit {\_f} +\mathit {\_F1} \left (-\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y \right )\] Contains RootOf