6.4.26 8.3

6.4.26.1 [1182] Problem 1

problem number 1182

Added March 10, 2019.

Problem Chapter 4.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.26.2 [1183] Problem 2

problem number 1183

Added March 10, 2019.

Problem Chapter 4.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = x f(\frac {y}{x}) w \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to e^{x f\left (\frac {y}{x}\right )} c_1\left (\frac {y}{x}\right )\right \}\right \}\]

Maple ✓

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

\[w \left (x , y\right ) = f_{1} \left (\frac {y}{x}\right ) {\mathrm e}^{f \left (\frac {y}{x}\right ) x}\]

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6.4.26.3 [1184] Problem 3

problem number 1184

Added March 10, 2019.

Problem Chapter 4.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = f(x^2+y^2) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x*D[w[x, y], x] + y*D[w[x, y], y] == f[x^2 + y^2]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  x*diff(w(x,y),x)+y*diff(w(x,y),y) = f(x^2+y^2)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.4.26.4 [1185] Problem 4

problem number 1185

Added March 10, 2019.

Problem Chapter 4.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = x^k f(x^n*y^m) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.26.5 [1186] Problem 5

problem number 1186

Added March 10, 2019.

Problem Chapter 4.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ m x w_x + n y w_y = f(a x^n+b y^m) w \]

Mathematica ✓

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

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

Maple ✓

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

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

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6.4.26.6 [1187] Problem 6

problem number 1187

Added March 10, 2019.

Problem Chapter 4.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + x y w_y = y^k f(\alpha x^n+\beta y^m) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  x^2*D[w[x, y], x] + x*y*D[w[x, y], y] == y^k*f[alpha*x + beta*y]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  x^2*diff(w(x,y),x)+x*y*diff(w(x,y),y) = y^k*f(alpha*x+beta*y)*w(x,y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y}{x}\right ) {\mathrm e}^{\int _{}^{x}\frac {\left (\frac {y \textit {\_a}}{x}\right )^{k} f \left (\frac {\textit {\_a} \left (\alpha x +\beta y \right )}{x}\right )}{\textit {\_a}^{2}}d \textit {\_a}}\]

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6.4.26.7 [1188] Problem 7

problem number 1188

Added March 10, 2019.

Problem Chapter 4.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ \frac {f(x)}{f'(x)} w_x + \frac {g(x)}{g'(x)} w_y = h(f(x)+g(y)) w \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  (f[x]*D[w[x, y], x])/Derivative[1][f][x] + (g[x]*D[w[x, y], y])/Derivative[1][g][x] == h[f[x] + g[y]]*w[x, y]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

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

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