6.2.22 7.3

6.2.22.1 [727] problem number 1

problem number 727

Added January 29, 2019.

Problem 2.7.3.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arctan(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'));
 

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

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6.2.22.2 [728] problem number 2

problem number 728

Added January 29, 2019.

Problem 2.7.3.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+( a*arctan(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'));
 

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

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6.2.22.3 [729] problem number 3

problem number 729

Added January 29, 2019.

Problem 2.7.3.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to \int _1^y\left (-\frac {b c_1}{b k \arctan (c+a x+b K[6860])^n+a}-\int _1^x\frac {a b^2 k n \arctan (c+a K[1]+b K[6860])^{n-1} c_1}{\left (b k \arctan (c+a K[1]+b K[6860])^n+a\right )^2 \left (c^2+2 a K[1] c+2 b K[6860] c+a^2 K[1]^2+b^2 K[6860]^2+2 a b K[1] K[6860]+1\right )}dK[1]\right )dK[6860]+\int _1^x\frac {b k \arctan (c+b y+a K[1])^n c_1}{b k \arctan (c+b y+a K[1])^n+a}dK[1]+c_2\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+k *arctan(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'));
 

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

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6.2.22.4 [730] problem number 4

problem number 730

Added January 29, 2019.

Problem 2.7.3.4 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.22.5 [731] problem number 5

problem number 731

Added January 29, 2019.

Problem 2.7.3.5 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (y^2 + lambda*ArcTan[x]^n*y - a^2 + a*lambda*ArcTan[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*arctan(x)^n*y -a^2 + a *lambda*arctan(x)^n )*diff(w(x,y),y) = 0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

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

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6.2.22.6 [732] problem number 6

problem number 732

Added January 29, 2019.

Problem 2.7.3.6 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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6.2.22.7 [733] problem number 7

problem number 733

Added Feb. 1, 2019.

Problem 2.7.3.7 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] - ((k + 1)*x^k*y^2 - lambda*ArcTan[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*arctan(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'));
 

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

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6.2.22.8 [734] problem number 8

problem number 734

Added Feb. 1, 2019.

Problem 2.7.3.8 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} \left (\left (\lambda -b^2 \lambda n\right ) \arctan (K[1])^n+a b\right )dK[1]\right )\right \}\right \}\]

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(lambda* arctan(x)^n +a*y+ a*b - b^2*lambda*arctan(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'));
 

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

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6.2.22.9 [735] problem number 9

problem number 735

Added Feb. 1, 2019.

Problem 2.7.3.9 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 - b*lambda*x^m*ArcTan[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*arctan(x)^n*y^2 - b*lambda*x^m*arctan(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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6.2.22.10 [736] problem number 10

problem number 736

Added Feb. 1, 2019.

Problem 2.7.3.10 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcTan[x]^n*y^2 + b*m*x^(m - 1) - lambda*b^2*x^(2*m)*ArcTan[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*arctan(x)^n*y^2 +b*m*x^(m-1) - lambda*b^2*x^(2*m)*arctan(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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6.2.22.11 [737] problem number 11

problem number 737

Added Feb. 1, 2019.

Problem 2.7.3.11 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + (lambda*ArcTan[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]];
 

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

Maple ✓

restart; 
pde :=  diff(w(x,y),x)+(lambda*arctan(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'));
 

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

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6.2.22.12 [738] problem number 12

problem number 738

Added Feb. 1, 2019.

Problem 2.7.3.12 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

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

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

Maple ✓

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

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

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