6.3.26 8.2

6.3.26.1 [998] Problem 1

problem number 998

Added Feb. 11, 2019.

Problem Chapter 3.8.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \frac {\int _{}^{x}\left (f \left (\textit {\_a} \right )+g \left (\frac {y a -b \left (x -\textit {\_a} \right )}{a}\right )\right )d \textit {\_a}}{a}+f_{1} \left (\frac {y a -b x}{a}\right )\]

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6.3.26.2 [999] Problem 2

problem number 999

Added Feb. 11, 2019.

Problem Chapter 3.8.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \int _{}^{x}f \left (\textit {\_a} \right ) g \left (\left (-x +\textit {\_a} \right ) a +y \right )d \textit {\_a} +f_{1} \left (-a x +y \right )\]

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6.3.26.3 [1000] Problem 3

problem number 1000

Added Feb. 11, 2019.

Problem Chapter 3.8.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + (a y+f(x) ) w_y = g(x) h(y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^xg(K[2]) h\left (e^{a K[2]} \left (e^{-a x} y-\int _1^xe^{-a K[1]} f(K[1])dK[1]+\int _1^{K[2]}e^{-a K[1]} f(K[1])dK[1]\right )\right )dK[2]+c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1]\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = \int _{}^{x}g \left (\textit {\_b} \right ) h \left (\left (\int f \left (\textit {\_b} \right ) {\mathrm e}^{-a \textit {\_b}}d \textit {\_b} -\int f \left (x \right ) {\mathrm e}^{-a x}d x +{\mathrm e}^{-a x} y \right ) {\mathrm e}^{a \textit {\_b}}\right )d \textit {\_b} +f_{1} \left (-\int f \left (x \right ) {\mathrm e}^{-a x}d x +{\mathrm e}^{-a x} y \right )\]

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6.3.26.4 [1001] Problem 4

problem number 1001

Added Feb. 11, 2019.

Problem Chapter 3.8.2.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f(x) w_x + g(y) w_y = h_1(x) + h_2(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  f[x]*D[w[x, y], x] + g[y]*D[w[x, y], y] == h1[x] + h2[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {c_1+\int _1^{K[6093]}\left (\text {h1}'(K[1])+\text {h2}'(K[1])\right )dK[1]}{f(K[6093])}dK[6093]+\int _1^y\frac {-c_1+\text {h1}(x)+\text {h2}(x)-\int _1^x\left (\text {h1}'(K[1])+\text {h2}'(K[1])\right )dK[1]}{g(K[6561])}dK[6561]+c_2\right \}\right \}\]

Maple ✓

restart; 
pde :=  f(x)*diff(w(x,y),x) +g(y)*diff(w(x,y),y) =   h1(x)+h2(y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {\operatorname {h1} \left (\textit {\_f} \right )+\operatorname {h2} \left (\operatorname {RootOf}\left (\int \frac {1}{f \left (\textit {\_f} \right )}d \textit {\_f} -\int _{}^{\textit {\_Z}}\frac {1}{g \left (\textit {\_a} \right )}d \textit {\_a} -\int \frac {1}{f \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y \right )\right )}{f \left (\textit {\_f} \right )}d \textit {\_f} +f_{1} \left (-\int \frac {1}{f \left (x \right )}d x +\int \frac {1}{g \left (y \right )}d y \right )\]

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6.3.26.5 [1002] Problem 5

problem number 1002

Added Feb. 11, 2019.

Problem Chapter 3.8.2.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) w_x + (f_2(x) y+y^k f_3(x)) w_y = g(x) h(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = f1[x]*D[w[x, y], x] + (y*f2[x] + y^k*f3[x])*D[w[x, y], y] == g[x]*h[x]; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {g(K[3]) h(K[3])}{\text {f1}(K[3])}dK[3]+c_1\left ((k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right ) \text {f3}(K[2])}{\text {f1}(K[2])}dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\frac {\text {f2}(K[1])}{\text {f1}(K[1])}dK[1]\right )\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f1(x)*diff(w(x,y),x) +(y*f2(x)+y^k*f3(x))*diff(w(x,y),y) =  g(x)*h(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int \frac {g \left (x \right ) h \left (x \right )}{\operatorname {f1} \left (x \right )}d x +f_{1} \left (\left (k -1\right ) \int \frac {\operatorname {f3} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x}}{\operatorname {f1} \left (x \right )}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x}\right )\]

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6.3.26.6 [1003] Problem 6

problem number 1003

Added Feb. 11, 2019.

Problem Chapter 3.8.2.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) g_1(x) w_x + f_2(x) g_2(x) w_y = h_1(x) h_2(x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = f1[x]*g1[x]*D[w[x, y], x] + f2[x]*g2[x]*D[w[x, y], y] == h1[x]*h2[x]; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^x\frac {\text {h1}(K[2]) \text {h2}(K[2])}{\text {f1}(K[2]) \text {g1}(K[2])}dK[2]+c_1\left (y-\int _1^x\frac {\text {f2}(K[1]) \text {g2}(K[1])}{\text {f1}(K[1]) \text {g1}(K[1])}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  f1(x)*g1(x)*diff(w(x,y),x) +f2(x)*g2(x)*diff(w(x,y),y) =  h1(x)*h2(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int \frac {\operatorname {h1} \left (x \right ) \operatorname {h2} \left (x \right )}{\operatorname {f1} \left (x \right ) \operatorname {g1} \left (x \right )}d x +f_{1} \left (-\int \frac {\operatorname {f2} \left (x \right ) \operatorname {g2} \left (x \right )}{\operatorname {f1} \left (x \right ) \operatorname {g1} \left (x \right )}d x +y \right )\]

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6.3.26.7 [1004] Problem 7

problem number 1004

Added Feb. 11, 2019.

Problem Chapter 3.8.2.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ f_1(x) g_1(y) w_x + f_2(x) g_2(y) w_y = h_1(x)+ h_2(x) \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = f1[x]*g1[y]*D[w[x, y], x] + f2[x]*g2[y]*D[w[x, y], y] == h1[x] + h2[x]; 
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) =  h1(x)+h2(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y)) ),output='realtime'));
 

\[w \left (x , y\right ) = \int _{}^{x}\frac {\operatorname {h1} \left (\textit {\_f} \right )+\operatorname {h2} \left (\textit {\_f} \right )}{\operatorname {f1} \left (\textit {\_f} \right ) \operatorname {g1} \left (\operatorname {RootOf}\left (\int \frac {\operatorname {f2} \left (\textit {\_f} \right )}{\operatorname {f1} \left (\textit {\_f} \right )}d \textit {\_f} -\int _{}^{\textit {\_Z}}\frac {\operatorname {g1} \left (\textit {\_a} \right )}{\operatorname {g2} \left (\textit {\_a} \right )}d \textit {\_a} -\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +\int \frac {\operatorname {g1} \left (y \right )}{\operatorname {g2} \left (y \right )}d y \right )\right )}d \textit {\_f} +f_{1} \left (-\int \frac {\operatorname {f2} \left (x \right )}{\operatorname {f1} \left (x \right )}d x +\int \frac {\operatorname {g1} \left (y \right )}{\operatorname {g2} \left (y \right )}d y \right )\]

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