6.5.20 7.2
6.5.20.1 [1328] Problem 1
problem number 1328
Added April 13, 2019.
Problem Chapter 5.7.2.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = w + c_1 \arccos ^k(\lambda x) + c_2 \arccos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == w[x,y]+ c1*ArcCos[lambda*x]^k+c2*ArcCos[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {x}{a}} \left (\int _1^x\frac {e^{-\frac {K[1]}{a}} \left (\text {c1} \arccos (\lambda K[1])^k+\text {c2} \arccos \left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = w(x,y)+c1*arccos(lambda*x)^k+c2*arccos(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}{\mathrm e}^{-\frac {\textit {\_a}}{a}} \left (\operatorname {c2} \arccos \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n}+\operatorname {c1} \arccos \left (\lambda \textit {\_a} \right )^{k}\right )d \textit {\_a}}{a}+f_{1} \left (\frac {a y -b x}{a}\right )\right ) {\mathrm e}^{\frac {x}{a}}\]
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6.5.20.2 [1329] Problem 2
problem number 1329
Added April 13, 2019.
Problem Chapter 5.7.2.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + \arccos ^k(\lambda x) \arccos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x,y]+ ArcCos[lambda*x]^k*ArcCos[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{\frac {c x}{a}} \left (\int _1^x\frac {e^{-\frac {c K[1]}{a}} \arccos (\lambda K[1])^k \arccos \left (\beta \left (y+\frac {b (K[1]-x)}{a}\right )\right )^n}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ arccos(lambda*x)^k*arccos(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\arccos \left (\lambda \textit {\_a} \right )^{k} \arccos \left (\frac {\left (y a -b \left (x -\textit {\_a} \right )\right ) \beta }{a}\right )^{n} {\mathrm e}^{-\frac {c \textit {\_a}}{a}}d \textit {\_a}}{a}+f_{1} \left (\frac {y a -b x}{a}\right )\right ) {\mathrm e}^{\frac {c x}{a}}\]
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6.5.20.3 [1330] Problem 3
problem number 1330
Added April 13, 2019.
Problem Chapter 5.7.2.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = \left ( c_1 \arccos (\lambda _1 x) + c_2 \arccos (\lambda _2 y)\right ) w+ s_1 \arccos ^n(\beta _1 x)+ s_2 \arccos ^k(\beta _2 y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == ( c1*ArcCos[lambda1*x] + c2*ArcCos[lambda2*y])*w[x,y]+ s1*ArcCos[beta1*x]^n+ s2*ArcCos[beta2*y]^k;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\frac {\text {c1} x \arccos (\text {lambda1} x)}{a}+\frac {\text {c2} x \arccos (\text {lambda2} y)}{a}+\frac {\text {c2} x \arcsin (\text {lambda2} y)}{a}-\frac {\text {c1} \sqrt {1-\text {lambda1}^2 x^2}}{a \text {lambda1}}-\frac {\text {c2} y \arcsin (\text {lambda2} y)}{b}-\frac {\text {c2} \sqrt {1-\text {lambda2}^2 y^2}}{b \text {lambda2}}\right ) \left (\int _1^x\frac {\exp \left (\frac {\text {c2} \text {lambda2} (a y-b x) \arcsin \left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) \text {lambda1}-b \text {c1} \text {lambda2} \arccos (\text {lambda1} K[1]) K[1] \text {lambda1}-b \text {c2} \text {lambda2} \arccos \left (\text {lambda2} \left (y+\frac {b (K[1]-x)}{a}\right )\right ) K[1] \text {lambda1}+a \text {c2} \sqrt {-y^2 \text {lambda2}^2-\frac {b^2 (x-K[1])^2 \text {lambda2}^2}{a^2}+\frac {2 b y (x-K[1]) \text {lambda2}^2}{a}+1} \text {lambda1}+b \text {c1} \text {lambda2} \sqrt {1-\text {lambda1}^2 K[1]^2}}{a b \text {lambda1} \text {lambda2}}\right ) \left (\text {s2} \arccos \left (\text {beta2} \left (y+\frac {b (K[1]-x)}{a}\right )\right )^k+\text {s1} \arccos (\text {beta1} K[1])^n\right )}{a}dK[1]+c_1\left (y-\frac {b x}{a}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*diff(w(x,y),y) = ( c1*arccos(lambda1*x) + c2*arccos(lambda2*y))*w(x,y)+ s1*arccos(beta1*x)^n+ s2*arccos(beta2*y)^k;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (f_{1} \left (\frac {a y -b x}{a}\right ) a +\int _{}^{x}{\mathrm e}^{\frac {\sqrt {-\frac {\left (\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} +a \right ) \left (\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} -a \right )}{a^{2}}}\, \operatorname {c2} a \lambda \operatorname {1} -\lambda \operatorname {2} \left (\operatorname {c2} \lambda \operatorname {1} \left (\left (\textit {\_a} -x \right ) b +a y \right ) \arccos \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \lambda \operatorname {2} }{a}\right )+b \operatorname {c1} \left (\arccos \left (\lambda \operatorname {1} \textit {\_a} \right ) \textit {\_a} \lambda \operatorname {1} -\sqrt {-\textit {\_a}^{2} \lambda \operatorname {1}^{2}+1}\right )\right )}{a \lambda \operatorname {1} \lambda \operatorname {2} b}} \left (\operatorname {s2} \arccos \left (\frac {\left (a y -b \left (x -\textit {\_a} \right )\right ) \beta \operatorname {2} }{a}\right )^{k}+\operatorname {s1} \arccos \left (\beta \operatorname {1} \textit {\_a} \right )^{n}\right )d \textit {\_a} \right ) {\mathrm e}^{\frac {-\sqrt {-\lambda \operatorname {1}^{2} x^{2}+1}\, \operatorname {c1} \lambda \operatorname {2} b +\lambda \operatorname {1} \left (-a \operatorname {c2} \sqrt {-y^{2} \lambda \operatorname {2}^{2}+1}+\lambda \operatorname {2} \left (a \arccos \left (\lambda \operatorname {2} y \right ) y \operatorname {c2} +\operatorname {c1} \arccos \left (\lambda \operatorname {1} x \right ) b x \right )\right )}{a \lambda \operatorname {1} \lambda \operatorname {2} b}}}{a}\]
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6.5.20.4 [1331] Problem 4
problem number 1331
Added April 13, 2019.
Problem Chapter 5.7.2.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu x) w + p \arccos ^n(\beta y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*ArcCos[mu*x]^m*D[w[x, y], y] == c*ArcCos[nu*x]^k*w[x,y]+p*ArcCos[beta*y]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \arccos (\nu K[2])^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \arccos (\nu K[2])^k}{a}dK[2]\right ) p \arccos \left (\beta \left (y-\int _1^x\frac {b \arccos (\mu K[1])^m}{a}dK[1]+\int _1^{K[3]}\frac {b \arccos (\mu K[1])^m}{a}dK[1]\right )\right ){}^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \arccos (\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*arccos(mu*x)^m*diff(w(x,y),y) = c*arccos(nu*x)^k*w(x,y)+p*arccos(beta*y)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[\text {Expression too large to display}\]
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6.5.20.5 [1332] Problem 5
problem number 1332
Added April 13, 2019.
Problem Chapter 5.7.2.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b \arccos ^m(\mu x) w_y = c \arccos ^k(\nu y) w + p \arccos ^n(\beta x) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*ArcCos[mu*x]^m*D[w[x, y], y] == c*ArcCos[nu*y]^k*w[x,y]+p*ArcCos[beta*x]^n;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {c \arccos \left (\nu \left (y-\int _1^x\frac {b \arccos (\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \arccos (\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {c \arccos \left (\nu \left (y-\int _1^x\frac {b \arccos (\mu K[1])^m}{a}dK[1]+\int _1^{K[2]}\frac {b \arccos (\mu K[1])^m}{a}dK[1]\right )\right ){}^k}{a}dK[2]\right ) p \arccos (\beta K[3])^n}{a}dK[3]+c_1\left (y-\int _1^x\frac {b \arccos (\mu K[1])^m}{a}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+ b*arccos(mu*x)^m*diff(w(x,y),y) = c*arccos(nu*y)^k*w(x,y)+p*arccos(beta*x)^n;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[\text {Expression too large to display}\]
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