6.5.24 8.2

6.5.24.1 [1355] Problem 1

problem number 1355

Added April 13, 2019.

Problem Chapter 5.8.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.24.2 [1356] Problem 2

problem number 1356

Added April 13, 2019.

Problem Chapter 5.8.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\left (\left (\textit {\_a} b +y a -x b \right ) g \left (\textit {\_a} \right )+\textit {\_a} f \left (\textit {\_a} \right ) a \right ) {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a^{2}}+f_{1} \left (\frac {y a -x b}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.24.3 [1357] Problem 3

problem number 1357

Added April 13, 2019.

Problem Chapter 5.8.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.24.4 [1358] Problem 4

problem number 1358

Added April 13, 2019.

Problem Chapter 5.8.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.24.5 [1359] Problem 5

problem number 1359

Added April 13, 2019.

Problem Chapter 5.8.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

6.5.24.6 [1360] Problem 6

problem number 1360

Added April 13, 2019.

Problem Chapter 5.8.2.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \frac {\text {f1}(x) \exp \left (\int _1^x\frac {a-\text {f1}'(K[1])}{\text {f1}(K[1])}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {a-\text {f1}'(K[1])}{\text {f1}(K[1])}dK[1]\right ) \text {g1}'(K[2])}{\text {f1}(K[2])}dK[2]+c_1\right )+\text {f2}(y) \exp \left (\int _1^y\frac {a-\text {f2}'(K[1])}{\text {f2}(K[1])}dK[1]\right ) \left (\int _1^y\frac {\exp \left (-\int _1^{K[2]}\frac {a-\text {f2}'(K[1])}{\text {f2}(K[1])}dK[1]\right ) \text {g2}'(K[2])}{\text {f2}(K[2])}dK[2]+c_2\right )-\text {g1}(x)-\text {g2}(y)}{a}\right \}\right \}\]

Maple ✓

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

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

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________