6.4.25 8.2

6.4.25.1 [1175] Problem 1

problem number 1175

Added March 10, 2019.

Problem Chapter 4.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.25.2 [1176] Problem 2

problem number 1176

Added March 10, 2019.

Problem Chapter 4.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.25.3 [1177] Problem 3

problem number 1177

Added March 10, 2019.

Problem Chapter 4.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.25.4 [1178] Problem 4

problem number 1178

Added March 10, 2019.

Problem Chapter 4.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{g \left (y \right )}d y \right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {h1} \left (\mathit {\_f} \right )+\mathit {h2} \left (\RootOf \left (\int \frac {1}{f \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {1}{f \left (x \right )}d x \right )+\int \frac {1}{g \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {1}{g \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{f \left (\mathit {\_f} \right )}d\mathit {\_f}}\] contains RootOf

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6.4.25.5 [1179] Problem 5

problem number 1179

Added March 10, 2019.

Problem Chapter 4.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

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

Failed

Maple ✗

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

sol=()

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6.4.25.6 [1180] Problem 6

problem number 1180

Added March 10, 2019.

Problem Chapter 4.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y \right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {h1} \left (\mathit {\_f} \right ) \mathit {h2} \left (\RootOf \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {g1} \left (\mathit {\_a} \right )}{\mathit {g2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{\mathit {f1} \left (\mathit {\_f} \right ) \mathit {g1} \left (\RootOf \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {g1} \left (\mathit {\_a} \right )}{\mathit {g2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d\mathit {\_f}}\] has RootOf

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6.4.25.7 [1181] Problem 7

problem number 1181

Added March 10, 2019.

Problem Chapter 4.8.2.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

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

\[w \left (x , y\right ) = \mathit {\_F1} \left (-\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y \right ) {\mathrm e}^{\int _{}^{x}\frac {\mathit {h1} \left (\mathit {\_f} \right )+\mathit {h2} \left (\RootOf \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {g1} \left (\mathit {\_a} \right )}{\mathit {g2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}{\mathit {f1} \left (\mathit {\_f} \right ) \mathit {g1} \left (\RootOf \left (\int \frac {\mathit {f2} \left (\mathit {\_f} \right )}{\mathit {f1} \left (\mathit {\_f} \right )}d \mathit {\_f} -\left (\int \frac {\mathit {f2} \left (x \right )}{\mathit {f1} \left (x \right )}d x \right )+\int \frac {\mathit {g1} \left (y \right )}{\mathit {g2} \left (y \right )}d y -\left (\int _{}^{\mathit {\_Z}}\frac {\mathit {g1} \left (\mathit {\_a} \right )}{\mathit {g2} \left (\mathit {\_a} \right )}d\mathit {\_a} \right )\right )\right )}d\mathit {\_f}}\] has RootOf

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