6.3.14 5.2

6.3.14.1 [930] Problem 1

problem number 930

Added Feb. 11, 2019.

Problem Chapter 3.5.2.1 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ a w_x + b w_y = c x^n+ s \ln ^k(\lambda y) \]

Mathematica ✓

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

\[\left \{\left \{w(x,y)\to c_1\left (y-\frac {b x}{a}\right )-\frac {x \left (-c x^n+n s+s\right )}{a (n+1)}+\frac {s y \log (\lambda y)}{b}\right \}\right \}\]

Maple ✓

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

\[w \left ( x,y \right ) ={\frac {1}{a \left ( 1+n \right ) b} \left ( \left ( 1+n \right ) \left ( {\it \_F1} \left ( {\frac {ya-xb}{a}} \right ) b+s \left ( \ln \left ( \lambda \,y \right ) -1 \right ) y \right ) a+c{x}^{1+n}b \right ) }\]

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6.3.14.2 [931] Problem 2

problem number 931

Added Feb. 11, 2019.

Problem Chapter 3.5.2.2 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b y^2+c x^n y+ s \ln ^k(\lambda x) \]

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x, y], x] + a*D[w[x, y], y] == b*y^2 + c*x^n*y + s*Log[lambda*x]^k; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
 

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!s \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{k}+c{{\it \_a}}^{1+n}a-c \left ( ax-y \right ) {{\it \_a}}^{n}+b \left ( \left ( x-{\it \_a} \right ) a-y \right ) ^{2}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \] Result has unresolved integrals

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6.3.14.3 [932] Problem 3

problem number 932

Added Feb. 11, 2019.

Problem Chapter 3.5.2.3 from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.

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

\[ w_x + a w_y = b ln^k(\lambda x) \ln ^n(\beta y) \]

Mathematica ✓

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

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

Maple ✓

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

\[w \left ( x,y \right ) =\int ^{x}\!b \left ( \ln \left ( {\it \_a}\,\lambda \right ) \right ) ^{k} \left ( \ln \left ( -\beta \, \left ( \left ( x-{\it \_a} \right ) a-y \right ) \right ) \right ) ^{n}{d{\it \_a}}+{\it \_F1} \left ( -ax+y \right ) \]

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