128 HFOPDE, chapter 5.2.4

____________________________________________________________________________________

128.1 Problem 1

problem number 1028

Added March 12, 2019.

Problem Chapter 5.2.4.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c w + k x^n y^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + k*x^n*y^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*diff(w(x,y),x)+ b*diff(w(x,y),y) = c*w(x,y)+ k*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_a}}^{n}k}{a} \left ( {\frac {b{\it \_a}+ya-bx}{a}} \right ) ^{m}{{\rm e}^{-{\frac {{\it \_a}\,c}{a}}}}}{d{\it \_a}}+{\it \_F1} \left ( {\frac {ya-bx}{a}} \right ) \right ) {{\rm e}^{{\frac {cx}{a}}}} \]

____________________________________________________________________________________

128.2 Problem 2

problem number 1029

Added March 12, 2019.

Problem Chapter 5.2.4.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + y w_y = b w + c x^n y^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*D[w[x, y], x] + y*D[w[x, y], y] == b*w[x, y] + c*x^n*y^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{\frac {x (b-m)}{a}} \left (e^{\frac {m x}{a}} c_1\left (y e^{-\frac {x}{a}}\right )-\frac {c y^m x^n \left (\frac {x (b-m)}{a}\right )^{-n} \text {Gamma}\left (n+1,\frac {x (b-m)}{a}\right )}{b-m}\right )\right \}\right \} \]

Maple ✗

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11); 
pde :=  a*diff(w(x,y),x)+ y*diff(w(x,y),y) = b*w(x,y)+ c*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text { Exception } \]

____________________________________________________________________________________

128.3 Problem 3

problem number 1030

Added April 1, 2019.

Problem Chapter 5.2.4.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

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

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*x*w[x, y] + b*x^n*y^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{a x} \left (c_1\left (\frac {y}{x}\right )-b y^m x^n (a x)^{-m-n} \text {Gamma}(m+n,a x)\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*x*w(x,y)+ b*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\frac { \left ( ax \right ) ^{-n/2-m/2} \WhittakerM \left ( n/2+m/2,m/2+n/2+1/2,ax \right ) {y}^{m}{x}^{n}b{{\rm e}^{1/2\,ax}}}{ \left ( n+m \right ) \left ( m+n+1 \right ) }}+{\frac { \WhittakerM \left ( n/2+m/2+1,m/2+n/2+1/2,ax \right ) \left ( ax \right ) ^{-n/2-m/2}{y}^{m}{x}^{n-1}b{{\rm e}^{1/2\,ax}}}{ \left ( n+m \right ) a}}+{{\rm e}^{ax}}{\it \_F1} \left ( {\frac {y}{x}} \right ) \]

____________________________________________________________________________________

128.4 Problem 4

problem number 1031

Added April 1, 2019.

Problem Chapter 5.2.4.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x w_x + y w_y = a \sqrt {x^2+y^2} w + b x^n y^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = x*D[w[x, y], x] + y*D[w[x, y], y] == a*Sqrt[x^2+y^2]*w[x,y] + b*x^n*y^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  x*diff(w(x,y),x)+ y*diff(w(x,y),y) = a*sqrt(x^2+y^2)*w(x,y)+ b*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( \left ( {\frac {a}{x}\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) }} \right ) ^{-m-n}{y}^{m}{x}^{-m}b \left ( {\frac {{x}^{n+m}}{ \left ( n+m \right ) \left ( m+n+1 \right ) } \left ( {\frac {a}{x}\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) }} \right ) ^{n+m} \left ( a\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) } \right ) ^{-n/2-m/2}{{\rm e}^{-1/2\,a\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) }}} \WhittakerM \left ( n/2+m/2,m/2+n/2+1/2,a\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) } \right ) }+{\frac {{x}^{n+m}}{ \left ( n+m \right ) a} \left ( {\frac {a}{x}\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) }} \right ) ^{n+m} \left ( a\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) } \right ) ^{-n/2-m/2}{{\rm e}^{-1/2\,a\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) }}} \WhittakerM \left ( n/2+m/2+1,m/2+n/2+1/2,a\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) } \right ) {\frac {1}{\sqrt {{x}^{2} \left ( {\frac {{y}^{2}}{{x}^{2}}}+1 \right ) }}}} \right ) +{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{a\sqrt {{x}^{2}+{y}^{2}}}} \]

____________________________________________________________________________________

128.5 Problem 5

problem number 1032

Added April 1, 2019.

Problem Chapter 5.2.4.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = c x^n y^m w + p x^k y^s \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*x^n*y^m*w[x,y] + p*x^k*y^s; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*x^n*y^m*w(x,y)+ p*x^k*y^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( {\frac {{y}^{s}p}{a} \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{-{k \left ( n+{\frac {bm}{a}} \right ) ^{-1}}-{\frac {sb}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}}{x}^{-{\frac {sb}{a}}} \left ( {\frac { \left ( an+bm \right ) ^{2}{y}^{-m}}{c \left ( ak+2\,an+2\,bm+sb \right ) \left ( ak+an+bm+sb \right ) \left ( ak+sb \right ) }{x}^{{kn \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {kbm}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {sbn}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {{b}^{2}ms}{{a}^{2}} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}-n} \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{{k \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {sb}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}} \left ( {\frac {c{x}^{n}{y}^{m}{a}^{2}{n}^{2}}{an+bm}}+2\,{\frac {c{x}^{n}{y}^{m}abmn}{an+bm}}+{\frac {c{x}^{n}{y}^{m}{b}^{2}{m}^{2}}{an+bm}}+{a}^{2}kn+{a}^{2}{n}^{2}+abkm+2\,abmn+abns+{b}^{2}{m}^{2}+{b}^{2}ms \right ) \left ( {\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) ^{-1/2\,{\frac {ak+an+bm+sb}{an+bm}}}{{\rm e}^{-1/2\,{\frac {c{x}^{n}{y}^{m}}{an+bm}}}} \WhittakerM \left ( {\frac {ak+sb}{an+bm}}-1/2\,{\frac {ak+an+bm+sb}{an+bm}},1/2\,{\frac {ak+an+bm+sb}{an+bm}}+1/2,{\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) }+{\frac { \left ( an+bm \right ) ^{2} \left ( ak+an+bm+sb \right ) {y}^{-m}}{ \left ( ak+sb \right ) \left ( ak+2\,an+2\,bm+sb \right ) c}{x}^{{kn \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {kbm}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {sbn}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {{b}^{2}ms}{{a}^{2}} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}-n} \left ( {\frac {{y}^{m}c}{an+bm}{x}^{-{\frac {bm}{a}}}} \right ) ^{{k \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\frac {sb}{a} \left ( n+{\frac {bm}{a}} \right ) ^{-1}}} \left ( {\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) ^{-1/2\,{\frac {ak+an+bm+sb}{an+bm}}}{{\rm e}^{-1/2\,{\frac {c{x}^{n}{y}^{m}}{an+bm}}}} \WhittakerM \left ( {\frac {ak+sb}{an+bm}}-1/2\,{\frac {ak+an+bm+sb}{an+bm}}+1,1/2\,{\frac {ak+an+bm+sb}{an+bm}}+1/2,{\frac {c{x}^{n}{y}^{m}}{an+bm}} \right ) } \right ) \left ( n+{\frac {bm}{a}} \right ) ^{-1}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {{\rm e}^{{\frac {c{x}^{n}{y}^{m}}{an+bm}}}} \]

____________________________________________________________________________________

128.6 Problem 6

problem number 1033

Added April 1, 2019.

Problem Chapter 5.2.4.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x w_x + b y w_y = (c x^n+ p y^m) w + q x^k y^s \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*x*D[w[x, y], x] + b*y*D[w[x, y], y] == c*(x^n+p*y^m)*w[x,y] + q*x^k*y^s; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

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

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*x*diff(w(x,y),x)+ b*y*diff(w(x,y),y) = c*(x^n+y^m)*w(x,y)+ q*x^k*y^s; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {{{\it \_b}}^{k-1}q}{a} \left ( y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) ^{s}{{\rm e}^{-{\frac {c}{a}\int \!{\frac {1}{{\it \_b}} \left ( {{\it \_b}}^{n}+ \left ( y{x}^{-{\frac {b}{a}}}{{\it \_b}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }\,{\rm d}{\it \_b}}}}}{d{\it \_b}}+{\it \_F1} \left ( y{x}^{-{\frac {b}{a}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {c}{{\it \_a}\,a} \left ( {{\it \_a}}^{n}+ \left ( y{x}^{-{\frac {b}{a}}}{{\it \_a}}^{{\frac {b}{a}}} \right ) ^{m} \right ) }{d{\it \_a}}}} \]

____________________________________________________________________________________

128.7 Problem 7

problem number 1034

Added April 1, 2019.

Problem Chapter 5.2.4.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + a x y w_y = b y^2 w + c x^n y^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = x^2*D[w[x, y], x] + a*x*y*D[w[x, y], y] == b*y^2*w[x,y] + c*x^n*y^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to e^{-\frac {b y^2}{x-2 a x}} \left (c_1\left (y x^{-a}\right )-\frac {c y^m x^{n-1} \left (-\frac {b y^2}{x-2 a x}\right )^{-\frac {a m+n-1}{2 a-1}} \text {Gamma}\left (\frac {a m+n-1}{2 a-1},-\frac {b y^2}{x-2 a x}\right )}{2 a-1}\right )\right \}\right \} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  x^2*diff(w(x,y),x)+ a*x*y*diff(w(x,y),y) = b*y^2*w(x,y)+ c*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) ={\it \_F1} \left ( y{x}^{-a} \right ) {{\rm e}^{{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}+{\frac {{y}^{m}c{x}^{-am}}{ \left ( 2\,a-1 \right ) {y}^{2}} \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{-{\frac {am+n-1}{2\,a-1}}} \left ( {\frac { \left ( 2\,a-1 \right ) ^{2} \left ( 2\,{a}^{2}m+4\,{a}^{2}-am+2\,an-6\,a-n+2 \right ) {x}^{am+n}}{b \left ( am+4\,a+n-3 \right ) \left ( am+n-1 \right ) \left ( am+2\,a+n-2 \right ) } \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{-1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}} \WhittakerM \left ( {\frac {am+n-1}{2\,a-1}}-1/2\,{\frac {am+2\,a+n-2}{2\,a-1}},1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}+1/2,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {am+n-1}{2\,a-1}}}{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}}+{\frac { \left ( 2\,a-1 \right ) ^{2} \left ( am+2\,a+n-2 \right ) {x}^{am+n}}{b \left ( am+4\,a+n-3 \right ) \left ( am+n-1 \right ) } \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{-1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}} \WhittakerM \left ( {\frac {am+n-1}{2\,a-1}}-1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}+1,1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}+1/2,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {am+n-1}{2\,a-1}}}{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}} \right ) {{\rm e}^{{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}}+{\frac {{y}^{m}c{x}^{-am} \left ( 2\,a-1 \right ) {x}^{am+n}}{b \left ( am+4\,a+n-3 \right ) \left ( am+n-1 \right ) \left ( am+2\,a+n-2 \right ) x} \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{-{\frac {am+n-1}{2\,a-1}}} \left ( {\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) ^{-1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}} \left ( 4\,{\frac {{a}^{2}b}{2\,a-1}}-4\,{\frac {ab}{2\,a-1}}+{\frac {b}{2\,a-1}} \right ) \WhittakerM \left ( {\frac {am+n-1}{2\,a-1}}-1/2\,{\frac {am+2\,a+n-2}{2\,a-1}},1/2\,{\frac {am+2\,a+n-2}{2\,a-1}}+1/2,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }} \right ) \left ( {\frac {b{y}^{2}{x}^{-2\,a}}{2\,a-1}} \right ) ^{{\frac {am+n-1}{2\,a-1}}}{{\rm e}^{-1/2\,{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}{{\rm e}^{{\frac {b{y}^{2}}{x \left ( 2\,a-1 \right ) }}}}} \]

____________________________________________________________________________________

128.8 Problem 8

problem number 1035

Added April 1, 2019.

Problem Chapter 5.2.4.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ x^2 w_x + x y w_y = y^2(a x+b y) w + c x^n y^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = x^2*D[w[x, y], x] + x*y*D[w[x, y], y] == y^2*(a*x+b*y)*w[x,y] + c*x^n*y^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to c_1\left (\frac {y}{x}\right ) e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )}-c 2^{\frac {1}{2} (m+n-3)} y^m x^{n-1} e^{\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )} \left (y^2 \left (a+\frac {b y}{x}\right )\right )^{\frac {1}{2} (-m-n+1)} \text {Gamma}\left (\frac {1}{2} (m+n-1),\frac {1}{2} y^2 \left (a+\frac {b y}{x}\right )\right )\right \}\right \} \]

Maple ✓

unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
 unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
 unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  x^2*diff(w(x,y),x)+ x*y*diff(w(x,y),y) = y^2*(a*x+b*y)*w(x,y)+ c*x^n*y^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( 1/4\,{2}^{m/2+n/2+1/2}c \left ( {\frac {by}{x}}+a \right ) {y}^{m+2} \left ( 4\,{\frac {{2}^{-n/4-m/4+3/4}{y}^{2}{x}^{n+m-1}}{ \left ( n+m-1 \right ) \left ( m+n+1 \right ) \left ( n+3+m \right ) } \WhittakerM \left ( m/4+n/4+1/4,n/4+m/4+3/4,1/2\,{y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) \left ( {\frac {by}{x}}+a \right ) \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) ^{-m/4-n/4-1/4} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-1/2+n/2+m/2}{{\rm e}^{-1/4\,{\frac {b{y}^{3}}{x}}-1/4\,{y}^{2}a}}}+4\,{\frac {{2}^{-n/4-m/4+3/4}{x}^{n+m-1}}{ \left ( m+n+1 \right ) \left ( n+m-1 \right ) {y}^{2}} \WhittakerM \left ( m/4+n/4+5/4,n/4+m/4+3/4,1/2\,{y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) +m+n+1 \right ) \left ( {y}^{2} \left ( {\frac {by}{x}}+a \right ) \right ) ^{-m/4-n/4-1/4} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-1/2+n/2+m/2}{{\rm e}^{-1/4\,{\frac {b{y}^{3}}{x}}-1/4\,{y}^{2}a}} \left ( {\frac {by}{x}}+a \right ) ^{-1}} \right ) {x}^{-m-2} \left ( {\frac {{y}^{2}}{{x}^{2}} \left ( {\frac {by}{x}}+a \right ) } \right ) ^{-m/2-n/2-1/2}+{\it \_F1} \left ( {\frac {y}{x}} \right ) \right ) {{\rm e}^{1/2\,{\frac {b{y}^{3}}{x}}+1/2\,{y}^{2}a}} \]

____________________________________________________________________________________

128.9 Problem 9

problem number 1036

Added April 1, 2019.

Problem Chapter 5.2.4.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b x^m y w_y = c x^p y^q w+s x^\gamma y^\delta + d \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*x^n*D[w[x, y], x] + b*x^m*y*D[w[x, y], y] == c*x^p*y^q*w[x,y] + s*x^gamma*y^delta+d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

Maple ✓

 
 unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
 unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
 unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*x^n*diff(w(x,y),x)+ b*x^m*y*diff(w(x,y),y) = c*x^p*y^q*w(x,y)+ s*x^gamma*y^delta+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ w \left ( x,y \right ) = \left ( \int ^{x}\!{\frac {1}{a}{{\rm e}^{-{\frac {c}{a}\int \!{{\it \_b}}^{-n+p} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}+{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) ^{q}\,{\rm d}{\it \_b}}}} \left ( {{\it \_b}}^{-n+\gamma }s \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}+{\frac {{{\it \_b}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) ^{\delta }+{{\it \_b}}^{-n}d \right ) }{d{\it \_b}}+{\it \_F1} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) \right ) {{\rm e}^{\int ^{x}\!{\frac {{{\it \_a}}^{-n+p}c}{a} \left ( y{{\rm e}^{-{\frac {{x}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}+{\frac {{{\it \_a}}^{-n+m+1}b}{a \left ( -n+m+1 \right ) }}}} \right ) ^{q}}{d{\it \_a}}}} \]

____________________________________________________________________________________

128.10 Problem 10

problem number 1037

Added April 1, 2019.

Problem Chapter 5.2.4.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + (b x^m y +c x^k ) w_y = s x^p y^q w+d \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*x^n*D[w[x, y], x] + (b*x^m*y+x*x^k)*D[w[x, y], y] == s*x^p*y^q*w[x,y] + d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

Maple ✓

unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
 unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
 unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*x^n*diff(w(x,y),x)+ (b*x^m*y+c*x^k)*diff(w(x,y),y) = s*x^p*y^q*w(x,y)+ d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
 

\[ \text {Too large to display} \]

____________________________________________________________________________________

128.11 Problem 11

problem number 1038

Added April 1, 2019.

Problem Chapter 5.2.4.11, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a x^n w_x + b x^m y^k w_y = c w + s x^p y^q + d \]

Mathematica ✗

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*x^n*D[w[x, y], x] + b*x^m*y^k*D[w[x, y], y] == c*w[x,y] + s*x^p*y^q+d; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \text {\$Aborted} \]

Maple ✓

unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
 unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
 unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*x^n*diff(w(x,y),x)+ b*x^m*y^k*diff(w(x,y),y) = c*w(x,y)+ s*x^p*y^q+d; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\frac {1}{a}{{\rm e}^{-{\frac {c{x}^{-n+1}}{a \left ( n-1 \right ) }}}} \left ( {\it \_F1} \left ( {\frac {b \left ( k-1 \right ) {x}^{-n+m+1}+{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) a+\int ^{x}\!{{\rm e}^{{\frac {c{{\it \_a}}^{-n+1}}{a \left ( n-1 \right ) }}}}{{\it \_a}}^{-n+p} \left ( \left ( {\frac {-b \left ( k-1 \right ) {{\it \_a}}^{-n+m+1}+b \left ( k-1 \right ) {x}^{-n+m+1}+{y}^{1-k}a \left ( -n+m+1 \right ) }{a \left ( -n+m+1 \right ) }} \right ) ^{- \left ( k-1 \right ) ^{-1}} \right ) ^{q}{d{\it \_a}}s+\int ^{x}\!{{\rm e}^{{\frac {c{{\it \_a}}^{-n+1}}{a \left ( n-1 \right ) }}}}{{\it \_a}}^{-n}{d{\it \_a}}d \right ) } \]

____________________________________________________________________________________

128.12 Problem 12

problem number 1039

Added April 1, 2019.

Problem Chapter 5.2.4.12, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a y^k w_x + b x^n w_y = c w + s x^m \]

Mathematica ✓

ClearAll[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 pde = a*y^k*D[w[x, y], x] + b*x^n*D[w[x, y], y] == c*w[x,y] + s*x^m; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]]; 
 sol = Simplify[sol];
 

\[ \left \{\left \{w(x,y)\to \exp \left (\frac {c x \left (\left (y^{-k-1}\right )^{-\frac {1}{k+1}}\right )^{-k} \left (\frac {a (n+1) y^{k+1}}{a (n+1) y^{k+1}-b (k+1) x^{n+1}}\right )^{\frac {k}{k+1}} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},\frac {1}{n+1}+1,\frac {b (k+1) x^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right ) \left (\int _1^x \frac {s K[1]^m \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k} \exp \left (-\frac {c K[1] \left (1-\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )^{\frac {k}{k+1}} \left (\left (\frac {a (n+1)}{a (n+1) y^{k+1}-b (k+1) \left (x^{n+1}-K[1]^{n+1}\right )}\right )^{-\frac {1}{k+1}}\right )^{-k} \text {Hypergeometric2F1}\left (\frac {k}{k+1},\frac {1}{n+1},\frac {1}{n+1}+1,\frac {b (k+1) K[1]^{n+1}}{b (k+1) x^{n+1}-a (n+1) y^{k+1}}\right )}{a}\right )}{a} \, dK[1]+c_1\left (\frac {y^{k+1}}{k+1}-\frac {b x^{n+1}}{a n+a}\right )\right )\right \}\right \} \]

Maple ✓

unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
 unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
 unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
pde :=  a*y^k*diff(w(x,y),x)+ b*x^n*diff(w(x,y),y) = c*w(x,y)+ s*x^m; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime')); 
sol:=simplify(sol);
 

\[ w \left ( x,y \right ) ={\frac {1}{a}{{\rm e}^{{\frac {c}{a}\int ^{x}\! \left ( \left ( {\frac {b \left ( k+1 \right ) {{\it \_a}}^{n+1}-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}{d{\it \_a}}}}} \left ( {\it \_F1} \left ( {\frac {-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) a+\int ^{x}\!{{\it \_a}}^{m} \left ( \left ( {\frac {b \left ( k+1 \right ) {{\it \_a}}^{n+1}-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}{{\rm e}^{-{\frac {c}{a}\int \! \left ( \left ( {\frac {b \left ( k+1 \right ) {{\it \_a}}^{n+1}-{x}^{n+1}b \left ( k+1 \right ) +{y}^{k+1}a \left ( n+1 \right ) }{a \left ( n+1 \right ) }} \right ) ^{ \left ( k+1 \right ) ^{-1}} \right ) ^{-k}\,{\rm d}{\it \_a}}}}{d{\it \_a}}s \right ) } \]