6.5.25 8.3
6.5.25.1 [1361] Problem 1
problem number 1361
Added April 13, 2019.
Problem Chapter 5.8.3.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ x w_x + y w_y = x f(\frac {y}{x}) w + g(x,y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = x*D[w[x, y], x] + y*D[w[x, y], y] == x*f[y/x]*w[x,y]+g[x,y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to e^{x f\left (\frac {y}{x}\right )} \left (\int _1^x\frac {e^{-f\left (\frac {y}{x}\right ) K[1]} g\left (K[1],\frac {y K[1]}{x}\right )}{K[1]}dK[1]+c_1\left (\frac {y}{x}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := x*diff(w(x,y),x)+ y*diff(w(x,y),y) = x*f(y/x)*w(x,y)+g(x,y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {g \left (\textit {\_a} , \frac {y \textit {\_a}}{x}\right ) {\mathrm e}^{-f \left (\frac {y}{x}\right ) \textit {\_a}}}{\textit {\_a}}d \textit {\_a} +f_{1} \left (\frac {y}{x}\right )\right ) {\mathrm e}^{f \left (\frac {y}{x}\right ) x}\]
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6.5.25.2 [1362] Problem 2
problem number 1362
Added April 13, 2019.
Problem Chapter 5.8.3.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a x w_x + b y w_y = f(x,y) w + g(x,y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == f[x,y]*w[x,y]+g[x,y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {f\left (K[1],x^{-\frac {b}{a}} y K[1]^{\frac {b}{a}}\right )}{a K[1]}dK[1]\right ) g\left (K[2],x^{-\frac {b}{a}} y K[2]^{\frac {b}{a}}\right )}{a K[2]}dK[2]+c_1\left (y x^{-\frac {b}{a}}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = f(x,y)*w(x,y)+g(x,y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\frac {\int _{}^{x}\frac {g \left (\textit {\_a} , y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right ) {\mathrm e}^{-\frac {\int \frac {f \left (\textit {\_a} , y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )}{\textit {\_a}}d \textit {\_a}}{a}}}{\textit {\_a}}d \textit {\_a}}{a}+f_{1} \left (y \,x^{-\frac {b}{a}}\right )\right ) {\mathrm e}^{\frac {\int _{}^{x}\frac {f \left (\textit {\_a} , y \,x^{-\frac {b}{a}} \textit {\_a}^{\frac {b}{a}}\right )}{\textit {\_a}}d \textit {\_a}}{a}}\]
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6.5.25.3 [1363] Problem 3
problem number 1363
Added April 13, 2019.
Problem Chapter 5.8.3.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + g(x) w_y = h(x,y) w + F(x,y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = f[x]*D[w[x, y], x] + g[x]*D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h\left (K[2],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[3]}\frac {h\left (K[2],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[2]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[2])}dK[2]\right ) F\left (K[3],y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]+\int _1^{K[3]}\frac {g(K[1])}{f(K[1])}dK[1]\right )}{f(K[3])}dK[3]+c_1\left (y-\int _1^x\frac {g(K[1])}{f(K[1])}dK[1]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := f(x)*diff(w(x,y),x)+ g(x)*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (\int _{}^{x}\frac {F \left (\textit {\_b} , \int \frac {g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} -\int \frac {g \left (x \right )}{f \left (x \right )}d x +y \right ) {\mathrm e}^{-\int \frac {h \left (\textit {\_b} , \int \frac {g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} -\int \frac {g \left (x \right )}{f \left (x \right )}d x +y \right )}{f \left (\textit {\_b} \right )}d \textit {\_b}}}{f \left (\textit {\_b} \right )}d \textit {\_b} +f_{1} \left (-\int \frac {g \left (x \right )}{f \left (x \right )}d x +y \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\textit {\_b} , \int \frac {g \left (\textit {\_b} \right )}{f \left (\textit {\_b} \right )}d \textit {\_b} -\int \frac {g \left (x \right )}{f \left (x \right )}d x +y \right )}{f \left (\textit {\_b} \right )}d \textit {\_b}}\]
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6.5.25.4 [1364] Problem 4
problem number 1364
Added April 13, 2019.
Problem Chapter 5.8.3.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y + g_0(x)) w_y = h(x,y) w + F(x,y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x])D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to \exp \left (\int _1^x\frac {h\left (K[3],\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {h\left (K[3],\exp \left (\int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[3]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[3])}dK[3]\right ) F\left (K[4],\exp \left (\int _1^{K[4]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+\int _1^{K[4]}\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )}{f(K[4])}dK[4]+c_1\left (y \exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )-\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x))*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,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 {F \left (\textit {\_f} , \left (\int \frac {\operatorname {g0} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} -\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +{\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x} y \right ) {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right ) {\mathrm e}^{-\int \frac {h \left (\textit {\_f} , \left (\int \frac {\operatorname {g0} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} -\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +{\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x} y \right ) {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} +f_{1} \left (-\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +{\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x} y \right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\textit {\_f} , \left (\int \frac {\operatorname {g0} \left (\textit {\_f} \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} -\int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +{\mathrm e}^{-\int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x} y \right ) {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\]
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6.5.25.5 [1365] Problem 5
problem number 1365
Added April 13, 2019.
Problem Chapter 5.8.3.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y + g_0(x) y^k) w_y = h(x,y) w + F(x,y) \]
Mathematica ✓
ClearAll["Global`*"];
pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*y^k)D[w[x, y], y] == h[x,y]*w[x,y]+F[x,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 {h\left (K[3],\left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]-(k-1) \int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k-\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^{K[3]}\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k+\exp \left (k \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )}{f(K[3])}dK[3]\right ) \left (\int _1^x\frac {\exp \left (-\int _1^{K[4]}\frac {h\left (K[3],\left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]-(k-1) \int _1^{K[3]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k-\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^{K[3]}\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k+\exp \left (k \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )}{f(K[3])}dK[3]\right ) F\left (K[4],\left (\exp \left (-\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]-(k-1) \int _1^{K[4]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y^{-k} \left (\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k-\exp \left (\int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) (k-1) \int _1^{K[4]}\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2] y^k+\exp \left (k \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) y\right )\right ){}^{\frac {1}{1-k}}\right )}{f(K[4])}dK[4]+c_1\left ((k-1) \int _1^x\frac {\exp \left ((k-1) \int _1^{K[2]}\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right ) \text {g0}(K[2])}{f(K[2])}dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\frac {\text {g1}(K[1])}{f(K[1])}dK[1]\right )\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*y^k)*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,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 {F \left (\textit {\_f} , \left (\left (1-k \right ) \int \frac {\operatorname {g0} \left (\textit {\_f} \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} +\left (k -1\right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right ) {\mathrm e}^{-\int \frac {h \left (\textit {\_f} , \left (\left (1-k \right ) \int \frac {\operatorname {g0} \left (\textit {\_f} \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} +\left (k -1\right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} +f_{1} \left (\left (k -1\right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}\right )\right ) {\mathrm e}^{\int _{}^{x}\frac {h \left (\textit {\_f} , \left (\left (1-k \right ) \int \frac {\operatorname {g0} \left (\textit {\_f} \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}}{f \left (\textit {\_f} \right )}d \textit {\_f} +\left (k -1\right ) \int \frac {\operatorname {g0} \left (x \right ) {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}}{f \left (x \right )}d x +y^{1-k} {\mathrm e}^{\left (k -1\right ) \int \frac {\operatorname {g1} \left (x \right )}{f \left (x \right )}d x}\right )^{-\frac {1}{k -1}} {\mathrm e}^{\int \frac {\operatorname {g1} \left (\textit {\_f} \right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\right )}{f \left (\textit {\_f} \right )}d \textit {\_f}}\]
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6.5.25.6 [1366] Problem 6
problem number 1366
Added April 13, 2019.
Problem Chapter 5.8.3.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ f(x) w_x + (g_1(x) y + g_0(x) e^{\lambda y}) w_y = h(x,y) w + F(x,y) \]
Mathematica ✗
ClearAll["Global`*"];
pde = f[x]*D[w[x, y], x] + (g1[x]*y+g0[x]*Exp[lambda*y])D[w[x, y], y] == h[x,y]*w[x,y]+F[x,y];
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
Failed
Maple ✗
restart;
pde := f(x)*diff(w(x,y),x)+ (g1(x)*y+g0(x)*exp(lambda*y))*diff(w(x,y),y) = h(x,y)*w(x,y)+F(x,y);
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
sol=()