6.2.23 7.4

6.2.23.1 [739] problem number 1

problem number 739

Added Feb. 1, 2019.

Problem 2.7.4.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arccot ^k(\lambda x)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCot[lambda*x]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (y-\int _1^x\left (\lambda \cot ^{-1}(\lambda K[1])^k+b\right )dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(lambda*arccot(lambda*x)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-b x +y -\lambda \int \operatorname {arccot}\left (\lambda x \right )^{k}d x \right )\]

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6.2.23.2 [740] problem number 2

problem number 740

Added Feb. 1, 2019.

Problem 2.7.4.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( a \arccot ^k(\lambda y)+b \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcCot[lambda*y]^k + b)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(lambda*arccot(lambda*y)^k+b)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\int \frac {1}{\lambda \operatorname {arccot}\left (\lambda y \right )^{k}+b}d y +x \right )\]

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6.2.23.3 [741] problem number 3

problem number 741

Added Feb. 1, 2019.

Problem 2.7.4.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arccot ^n(a x+b y+c) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + k*ArcCot[a*x + b*y + c]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {b c_1}{b k \cot ^{-1}(c+a x+b K[6326])^n+a}-\int _1^x-\frac {a b^2 k n \cot ^{-1}(c+a K[1]+b K[6326])^{n-1} c_1}{\left (b k \cot ^{-1}(c+a K[1]+b K[6326])^n+a\right )^2 \left (c^2+2 a K[1] c+2 b K[6326] c+a^2 K[1]^2+b^2 K[6326]^2+2 a b K[1] K[6326]+1\right )}dK[1]\right )dK[6326]+\int _1^x\frac {b k \cot ^{-1}(c+b y+a K[1])^n c_1}{b k \cot ^{-1}(c+b y+a K[1])^n+a}dK[1]+c_2\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+k*arccot(a*x+b*y+c)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.2.23.4 [742] problem number 4

problem number 742

Added Feb. 1, 2019.

Problem 2.7.4.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + k \arccot ^k(\lambda x) \arccot ^n(\mu y) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*ArcCot[lambda*x]^k*ArcCot[lambda*y]^n*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^y\cot ^{-1}(\lambda K[1])^{-n}dK[1]-\int _1^xa \cot ^{-1}(\lambda K[2])^kdK[2]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+a*arccot(lambda*x)^k*arccot(lambda*y)^n*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (-\int \operatorname {arccot}\left (\lambda x \right )^{k}d x +\frac {\int \operatorname {arccot}\left (\lambda y \right )^{-n}d y}{a}\right )\]

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6.2.23.5 [743] problem number 5

problem number 743

Added Feb. 1, 2019.

Problem 2.7.4.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+ \lambda (\arccot x)^n y - a^2 +a \lambda (\arccot x)^n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*ArcCot[x]^n*y - a^2 + a*lambda*ArcCot[x]^n)*D[w[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(y^2+lambda*arccot(x)^n*y - a^2 +a*lambda*arccot(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-{\mathrm e}^{-\int \left (-\operatorname {arccot}\left (x \right )^{n} \lambda +2 a \right )d x}+\int {\mathrm e}^{-\int \left (-\operatorname {arccot}\left (x \right )^{n} \lambda +2 a \right )d x}d x \left (-a -y \right )}{a +y}\right )\]

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6.2.23.6 [744] problem number 6

problem number 744

Added Feb. 1, 2019.

Problem 2.7.4.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( y^2+ \lambda x (\arccot x)^n y + \lambda (\arccot x)^n \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[w[x, y], x] + (y^2 + lambda*x*ArcCot[x]^n*y + lambda*ArcCot[x]^n)*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(y^2+lambda*x*arccot(x)^n*y +lambda*arccot(x)^n)*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {y x \int {\mathrm e}^{\int \frac {\lambda \operatorname {arccot}\left (x \right )^{n} x^{2}-2}{x}d x}d x +{\mathrm e}^{\int \frac {\lambda \operatorname {arccot}\left (x \right )^{n} x^{2}-2}{x}d x} x +\int {\mathrm e}^{\int \frac {\lambda \operatorname {arccot}\left (x \right )^{n} x^{2}-2}{x}d x}d x}{y x +1}\right )\]

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6.2.23.7 [745] problem number 7

problem number 745

Added Feb. 1, 2019.

Problem 2.7.4.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x - \left ( (k+1) x^k y^2- \lambda (\arccot x)^n (x^{k+1} y -1) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcCot[x]^n*(x^(k + 1)*y - 1))*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,y),x)-((k+1)*x^k*y^2- lambda*arccot(x)^n*(x^(k+1)*y-1))*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-x^{k +1} {\mathrm e}^{\int \frac {\operatorname {arccot}\left (x \right )^{n} x^{k +1} x \lambda -2 k -2}{x}d x}+\int x^{-k -2} {\mathrm e}^{\lambda \int x^{k +1} \operatorname {arccot}\left (x \right )^{n}d x}d x \left (x^{k +1} y -1\right ) \left (k +1\right )}{x^{k +1} y -1}\right )\]

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6.2.23.8 [746] problem number 8

problem number 746

Added Feb. 1, 2019.

Problem 2.7.4.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n y^2+a y + a b -b^2 \lambda (\arccot x)^n n \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 + a*y + a*b - b^2*lambda*ArcCot[x]^n*n)*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+(lambda*arccot(x)^n*y^2+a*y + a*b -b^2*lambda*arccot(x)^n*n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.23.9 [747] problem number 9

problem number 747

Added Feb. 1, 2019.

Problem 2.7.4.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n y^2- b \lambda x^m(\arccot x)^n y+ b m x^{m-1} \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 - b*lambda*x^m*ArcCot[x]^n*y + b*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+(lambda*arccot(x)^n*y^2- b*lambda*x^m*arccot(x)^n*y+ b*m*x^(m-1)  )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.23.10 [748] problem number 10

problem number 748

Added Feb. 1, 2019.

Problem 2.7.4.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n y^2+ b m x^{m-1} - \lambda b^2 x^{2 m} (\arccot x^n) \right ) w_y = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcCot[x]^n)*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccot(x)^n*y^2+ b*m*x^(m-1) - lambda*b^2*x^(2*m)*arccot(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

sol=()

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6.2.23.11 [749] problem number 11

problem number 749

Added Feb. 1, 2019.

Problem 2.7.4.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + \left ( \lambda (\arccot x)^n(y-a x^m-b)^2+a m x^{m-1} \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = D[w[x, y], x] + (lambda*ArcCot[x]^n*(y - a*x^m - b)^2 + a*m*x^(m - 1))*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\int _1^x\lambda \cot ^{-1}(K[2])^ndK[2]-\frac {1}{a x^m+b-y}\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( lambda*arccot(x)^n*(y-a*x^m-b)^2+a*m*x^(m-1) )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (\frac {-1+\lambda \left (a \,x^{m}+b -y \right ) \int \operatorname {arccot}\left (x \right )^{n}d x}{a \,x^{m}+b -y}\right )\]

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6.2.23.12 [750] problem number 12

problem number 750

Added Feb. 1, 2019.

Problem 2.7.4.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + \left ( \lambda (\arccot x)^n y^2+ k y+ \lambda b^2 x^{2 k} (\arccot x)^n \right ) w_y = 0 \]

Mathematica ✓

ClearAll["Global`*"]; 
pde = x*D[w[x, y], x] + (lambda*ArcCot[x]^n*y^2 + k*y + lambda*b^2*x^(2*k)*ArcCot[x]^n)*D[w[x, y], y] == 0; 
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

\[\left \{\left \{w(x,y)\to c_1\left (\arctan \left (\frac {y x^{-k}}{\sqrt {b^2}}\right )-\sqrt {b^2} \int _1^x\lambda \cot ^{-1}(K[1])^n K[1]^{k-1}dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  x*diff(w(x,y),x)+( lambda*arccot(x)^n*y^2+ k*y+ lambda*b^2*x^(2*k)*arccot(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[w \left (x , y\right ) = f_{1} \left (b \lambda \int x^{k -1} \operatorname {arccot}\left (x \right )^{n}d x -\arctan \left (\frac {x^{-k} y}{b}\right )\right )\]

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