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'));
 

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {1}{a} \left ( f \left ( {\it \_a} \right ) +g \left ( {\frac {ay-b \left ( x-{\it \_a} \right ) }{a}} \right ) \right ) }{d{\it \_a}}+{\it \_F1} \left ( {\frac {ay-xb}{a}} \right ) \]

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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'));
 

\[w \left ( x,y \right ) =\int ^{x}\!f \left ( {\it \_a} \right ) g \left ( \left ( -x+{\it \_a} \right ) a+y \right ) {d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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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'));
 

\[w \left ( x,y \right ) =\int ^{x}\!g \left ( {\it \_b} \right ) h \left ( \left ( \int \!f \left ( {\it \_b} \right ) {{\rm e}^{-a{\it \_b}}}\,{\rm d}{\it \_b}-\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) {{\rm e}^{a{\it \_b}}} \right ) {d{\it \_b}}+{\it \_F1} \left ( -\int \!f \left ( x \right ) {{\rm e}^{-ax}}\,{\rm d}x+y{{\rm e}^{-ax}} \right ) \]

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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]];
 

Failed

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'));
 

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {{\it h1} \left ( {\it \_f} \right ) +{\it h2} \left ( \RootOf \left ( \int \! \left ( f \left ( {\it \_f} \right ) \right ) ^{-1}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\! \left ( g \left ( {\it \_a} \right ) \right ) ^{-1}{d{\it \_a}}-\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \right ) }{f \left ( {\it \_f} \right ) }}{d{\it \_f}}+{\it \_F1} \left ( -\int \! \left ( f \left ( x \right ) \right ) ^{-1}\,{\rm d}x+\int \! \left ( g \left ( y \right ) \right ) ^{-1}\,{\rm d}y \right ) \]

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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'));
 

\[w \left ( x,y \right ) =\int \!{\frac {g \left ( x \right ) h \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( \left ( k-1 \right ) \int \!{\frac {{\it f3} \left ( x \right ) }{{\it f1} \left ( x \right ) }{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x}}}\,{\rm d}x+{y}^{-k+1}{{\rm e}^{ \left ( k-1 \right ) \int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x}} \right ) \]

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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'));
 

\[w \left ( x,y \right ) =\int \!{\frac {{\it h1} \left ( x \right ) {\it h2} \left ( x \right ) }{{\it f1} \left ( x \right ) {\it g1} \left ( x \right ) }}\,{\rm d}x+{\it \_F1} \left ( -\int \!{\frac {{\it f2} \left ( x \right ) {\it g2} \left ( x \right ) }{{\it f1} \left ( x \right ) {\it g1} \left ( x \right ) }}\,{\rm d}x+y \right ) \]

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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'));
 

\[w \left ( x,y \right ) =\int ^{x}\!{\frac {{\it h1} \left ( {\it \_f} \right ) +{\it h2} \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) } \left ( {\it g1} \left ( \RootOf \left ( \int \!{\frac {{\it f2} \left ( {\it \_f} \right ) }{{\it f1} \left ( {\it \_f} \right ) }}\,{\rm d}{\it \_f}-\int ^{{\it \_Z}}\!{\frac {{\it g1} \left ( {\it \_a} \right ) }{{\it g2} \left ( {\it \_a} \right ) }}{d{\it \_a}}-\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ) \right ) \right ) ^{-1}}{d{\it \_f}}+{\it \_F1} \left ( -\int \!{\frac {{\it f2} \left ( x \right ) }{{\it f1} \left ( x \right ) }}\,{\rm d}x+\int \!{\frac {{\it g1} \left ( y \right ) }{{\it g2} \left ( y \right ) }}\,{\rm d}y \right ) \]

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