7.3.14 5.2

7.3.14.1 [932] Problem 1

problem number 932

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]];
 

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

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'));
 

$$w(x,y)=\frac{bc{x}^{n+1}+(n+1)(b\mathit{\text{\_F1}}\left(\frac{ay-bx}{a}\right)+(\ln \left(\lambda y\right)-1)sy)a}{(n+1)ab}$$

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7.3.14.2 [933] Problem 2

problem number 933

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]];
 

$$\left\{\{w(x,y)\to \frac{(-\log (\lambda x){)}^{-k}(3(n^2+3n+2)s{\log }^k(\lambda x)\mathrm{Gamma}(k+1,-\log (\lambda x))+3\lambda (n^2+3n+2)(-\log (\lambda x){)}^k{c}_1(y-ax)+\lambda x(-\log (\lambda x){)}^k(a^{2}b(n^2+3n+2)x^2-3ax(b(n^2+3n+2)y+c{x}^n)+3(n+2)y(b(n+1)y+c{x}^n)))}{3\lambda (n+1)(n+2)}\}\right\}$$

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'));
 

$$w(x,y)={\int }_{}^x(ac{\mathit{\text{\_a}}}^{n+1}-(ax-y)c{\mathit{\text{\_a}}}^n+s\ln {\left(\mathit{\text{\_a}}\lambda \right)}^k+{((-\mathit{\text{\_a}}+x)a-y)}^{2}b)d\mathit{\text{\_a}}+\mathit{\text{\_F1}}(-ax+y)$$ Result has unresolved integrals

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7.3.14.3 [934] Problem 3

problem number 934

Added Feb. 11, 2019.

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

Solve for $w(x,y)$

$$w_x+a{w}_y=bl{n}^k(\lambda x){\ln }^n(\beta y)$$

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]];
 

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

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'));
 

$$w(x,y)={\int }_{}^{x}b\ln {\left(\mathit{\text{\_a}}\lambda \right)}^k\ln {(-((-\mathit{\text{\_a}}+x)a-y)\beta )}^{n}d\mathit{\text{\_a}}+\mathit{\text{\_F1}}(-ax+y)$$

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