6.5.1 2.1
6.5.1.1 [1196] Problem 1
problem number 1196
Added March 10, 2019.
Problem Chapter 5.2.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a w_x + b w_y = c w + d \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*D[w[x, y], x] + b*D[w[x, y], y] == c*w[x, y] + d;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {d}{c}+e^{\frac {c x}{a}} c_1\left (y-\frac {b x}{a}\right )\right \}\right \}\]
Maple ✓
restart;
pde := a*diff(w(x,y),x)+b*diff(w(x,y),y) = c*w(x,y)+d;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {{\mathrm e}^{\frac {c x}{a}} f_{1} \left (\frac {a y -b x}{a}\right ) c -d}{c}\]
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6.5.1.2 [1197] Problem 2
problem number 1197
Added March 10, 2019.
Problem Chapter 5.2.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (x-a) w_x + (y-b) w_y = w - c \]
Mathematica ✓
ClearAll["Global`*"];
pde = (x - a)*D[w[x, y], x] + (y - b)*D[w[x, y], y] == w[x, y] - c;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to c+(x-a) c_1\left (\frac {b-y}{a-x}\right )\right \}\right \}\]
Maple ✓
restart;
pde := (x-a)*diff(w(x,y),x)+(y-b)*diff(w(x,y),y) = w(x,y)-c;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \left (-x +a \right ) f_{1} \left (\frac {y -b}{-x +a}\right )+c\]
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6.5.1.3 [1198] Problem 3
problem number 1198
Added March 10, 2019.
Problem Chapter 5.2.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c x+d) w_y = \alpha w + \beta \]
Mathematica ✓
ClearAll["Global`*"];
pde = (a*x + b)*D[w[x, y], x] + (c*x + d)*D[w[x, y], y] == alpha*w[x, y] + beta;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\beta }{\alpha }+(a x+b)^{\frac {\alpha }{a}} c_1\left (y-\int _1^x\frac {d+c K[1]}{b+a K[1]}dK[1]\right )\right \}\right \}\]
Maple ✓
restart;
pde := (a*x+b)*diff(w(x,y),x)+ (c*x+d)*diff(w(x,y),y) = alpha*w(x,y)+beta;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (a x +b \right )^{\frac {\alpha }{a}} f_{1} \left (\frac {\left (-a d +b c \right ) \ln \left (a x +b \right )+y \,a^{2}-c x a}{a^{2}}\right ) \alpha -\beta }{\alpha }\]
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6.5.1.4 [1199] Problem 4
problem number 1199
Added March 10, 2019.
Problem Chapter 5.2.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c y+d) w_y = \alpha w + \beta \]
Mathematica ✓
ClearAll["Global`*"];
pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*w[x, y] + beta;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\beta }{\alpha }+(a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )\right \}\right \}\]
Maple ✓
restart;
pde := (a*x+b)*diff(w(x,y),x)+ (c*y+d)*diff(w(x,y),y) = alpha*w(x,y)+beta;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (a x +b \right )^{\frac {\alpha }{a}} f_{1} \left (\frac {\left (c y +d \right ) \left (a x +b \right )^{-\frac {c}{a}}}{c}\right ) \alpha -\beta }{\alpha }\]
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6.5.1.5 [1200] Problem 5
problem number 1200
Added March 10, 2019.
Problem Chapter 5.2.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c y+d) w_y = \alpha w + \beta y+\gamma x \]
Mathematica ✓
ClearAll["Global`*"];
pde = (a*x + b)*D[w[x, y], x] + (c*y + d)*D[w[x, y], y] == alpha*w[x, y] + beta*y + gamma*x;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\alpha (a-\alpha ) (\alpha -c) (a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(c y+d) (a x+b)^{-\frac {c}{a}}}{c}\right )-a \beta (\alpha y+d)+\alpha ^2 \beta y+\alpha ^2 \gamma x+\alpha b \gamma +\alpha \beta d-\alpha c \gamma x-b c \gamma }{\alpha (\alpha -a) (\alpha -c)}\right \}\right \}\]
Maple ✓
restart;
pde := (a*x+b)*diff(w(x,y),x)+ (c*y+d)*diff(w(x,y),y) = alpha*w(x,y)+beta*y+gamma*x;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (a x +b \right )^{\frac {\alpha }{a}} \alpha \left (c -\alpha \right ) \left (a -\alpha \right ) f_{1} \left (\frac {\left (c y +d \right ) \left (a x +b \right )^{-\frac {c}{a}}}{c}\right )+\left (-\beta y -\gamma x \right ) \alpha ^{2}+\left (a \beta y +\gamma x c -b \gamma -\beta d \right ) \alpha +a \beta d +b c \gamma }{\alpha \left (a -\alpha \right ) \left (c -\alpha \right )}\]
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6.5.1.6 [1201] Problem 6
problem number 1201
Added March 10, 2019.
Problem Chapter 5.2.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a x+b) w_x + (c x+d y) w_y = \alpha w + \beta \]
Mathematica ✓
ClearAll["Global`*"];
pde = (a*x + b)*D[w[x, y], x] + (c*x + d*y)*D[w[x, y], y] == alpha*w[x, y] + beta;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to -\frac {\beta }{\alpha }+(a x+b)^{\frac {\alpha }{a}} c_1\left (\frac {(a x+b)^{-\frac {d}{a}} (-d (-a y+c x+d y)-b c)}{d (a-d)}\right )\right \}\right \}\]
Maple ✓
restart;
pde := (a*x+b)*diff(w(x,y),x)+ (c*x+d*y)*diff(w(x,y),y) = alpha*w(x,y)+beta;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (a x +b \right )^{\frac {\alpha }{a}} f_{1} \left (\frac {\left (d y a -c x d -d^{2} y -b c \right ) \left (a x +b \right )^{-\frac {d}{a}}}{d \left (a -d \right )}\right ) \alpha -\beta }{\alpha }\]
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6.5.1.7 [1202] Problem 7
problem number 1202
Added March 10, 2019.
Problem Chapter 5.2.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ (a_1 x+a_0) w_x + (b_2 y+b_1 x+b_0) w_y = (c_2 y+c_1 x+c_0) w + k_2 y+k_1 x+k_0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = (a1*x + a0)*D[w[x, y], x] + (b2*y + b1*x + b0)*D[w[x, y], y] == (c2*y + c1*x + c0)*w[x, y] + k2*y + k1*x + k0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\[\left \{\left \{w(x,y)\to (\text {a0}+\text {a1} x)^{\frac {\text {a0} \text {b1} \text {c2}-\text {a0} \text {b2} \text {c1}-\text {a1} \text {b0} \text {c2}+\text {a1} \text {b2} \text {c0}}{\text {a1}^2 \text {b2}}} \exp \left (\frac {-\text {a1} (\text {a0} \text {b1} \text {c2}+\text {b2} (\text {b0} \text {c2}+\text {b1} \text {c2} x-\text {b2} \text {c1} x+\text {b2} \text {c2} y))+\text {a1}^2 \text {c2} (\text {b0}+\text {b2} y)+\text {b2}^2 x (\text {b1} \text {c2}-\text {b2} \text {c1})}{\text {a1} \text {b2}^2 (\text {a1}-\text {b2})}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\frac {(\text {a1} \text {c1}-\text {b2} \text {c1}+\text {b1} \text {c2}) K[1]}{\text {a1}}-\frac {\text {c2} (\text {a0}+\text {a1} x)^{-\frac {\text {b2}}{\text {a1}}} (\text {a0} \text {b1}-\text {a1} (\text {b0}+\text {b2} y)+\text {b2} (\text {b0}+\text {b1} x+\text {b2} y)) (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}}{\text {b2}^2}}{\text {a1}-\text {b2}}\right ) (\text {a0}+\text {a1} x)^{-\frac {\text {b2}}{\text {a1}}} (\text {a0}+\text {a1} K[1])^{\frac {-\text {a1} \text {b2} \text {c0}+\text {a0} \text {b2} \text {c1}+\text {a1} \text {b0} \text {c2}-\text {a0} \text {b1} \text {c2}}{\text {a1}^2 \text {b2}}-1} \left (-\left (\left (\text {k0} (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k1} K[1] (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k2} y (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right ) \text {b2}^2\right )+\text {a1} \left (\text {k0} (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k1} K[1] (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}+\text {k2} y (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right ) \text {b2}+\text {k2} \left (\text {b1} K[1] (\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}-\text {b1} x (\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}+\text {b0} \left ((\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}-(\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right )\right ) \text {b2}+\text {a0} \text {b1} \text {k2} \left ((\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}-(\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}\right )+\text {a1} \text {b0} \text {k2} \left ((\text {a0}+\text {a1} K[1])^{\frac {\text {b2}}{\text {a1}}}-(\text {a0}+\text {a1} x)^{\frac {\text {b2}}{\text {a1}}}\right )\right )}{(\text {a1}-\text {b2}) \text {b2}}dK[1]+c_1\left (\frac {(\text {a0}+\text {a1} x)^{-\frac {\text {b2}}{\text {a1}}} (-\text {a0} \text {b1}+\text {a1} (\text {b0}+\text {b2} y)-\text {b2} (\text {b0}+\text {b1} x+\text {b2} y))}{\text {b2} (\text {a1}-\text {b2})}\right )\right )\right \}\right \}\]
Maple ✓
restart;
pde := (a1*x+a0)*diff(w(x,y),x)+ (b2*y+b1*x+b0)*diff(w(x,y),y) = (c2*y+c1*x+c0)*w(x,y)+k2*y+k1*x+k0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = \frac {\left (-\int _{}^{x}{\mathrm e}^{\frac {\left (\textit {\_a} \operatorname {a1} +\operatorname {a0} \right )^{\frac {\operatorname {b2}}{\operatorname {a1}}} \left (\operatorname {b2}^{2} y +\left (-\operatorname {a1} y +\operatorname {b1} x +\operatorname {b0} \right ) \operatorname {b2} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} \right ) \operatorname {a1} \operatorname {c2} \left (\operatorname {a1} x +\operatorname {a0} \right )^{-\frac {\operatorname {b2}}{\operatorname {a1}}}-\textit {\_a} \,\operatorname {b2}^{2} \left (\operatorname {c1} \operatorname {a1} +\operatorname {b1} \operatorname {c2} -\operatorname {c1} \operatorname {b2} \right )}{\left (\operatorname {a1} -\operatorname {b2} \right ) \operatorname {b2}^{2} \operatorname {a1}}} \left (\left (\operatorname {b2}^{2} y +\left (-\operatorname {a1} y +\operatorname {b1} x +\operatorname {b0} \right ) \operatorname {b2} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} \right ) \operatorname {k2} \left (\operatorname {a1} x +\operatorname {a0} \right )^{-\frac {\operatorname {b2}}{\operatorname {a1}}} \left (\textit {\_a} \operatorname {a1} +\operatorname {a0} \right )^{\frac {-\operatorname {a1}^{2} \operatorname {b2} +\left (\operatorname {b0} \operatorname {c2} +\operatorname {b2}^{2}-\operatorname {b2} \operatorname {c0} \right ) \operatorname {a1} -\operatorname {a0} \left (\operatorname {b1} \operatorname {c2} -\operatorname {c1} \operatorname {b2} \right )}{\operatorname {a1}^{2} \operatorname {b2}}}-\left (\left (-\textit {\_a} \operatorname {k1} -\operatorname {k0} \right ) \operatorname {b2}^{2}+\left (\left (\textit {\_a} \operatorname {b1} +\operatorname {b0} \right ) \operatorname {k2} +\operatorname {a1} \left (\textit {\_a} \operatorname {k1} +\operatorname {k0} \right )\right ) \operatorname {b2} +\operatorname {k2} \left (\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} \right )\right ) \left (\textit {\_a} \operatorname {a1} +\operatorname {a0} \right )^{\frac {\left (\operatorname {a0} \operatorname {c1} -\operatorname {a1}^{2}-\operatorname {a1} \operatorname {c0} \right ) \operatorname {b2} -\operatorname {c2} \left (\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} \right )}{\operatorname {a1}^{2} \operatorname {b2}}}\right )d \textit {\_a} +f_{1} \left (-\frac {\left (\operatorname {b2}^{2} y +\left (-\operatorname {a1} y +\operatorname {b1} x +\operatorname {b0} \right ) \operatorname {b2} +\operatorname {a0} \operatorname {b1} -\operatorname {a1} \operatorname {b0} \right ) \left (\operatorname {a1} x +\operatorname {a0} \right )^{-\frac {\operatorname {b2}}{\operatorname {a1}}}}{\left (\operatorname {a1} -\operatorname {b2} \right ) \operatorname {b2}}\right ) \operatorname {b2} \left (\operatorname {a1} -\operatorname {b2} \right )\right ) \left (\operatorname {a1} x +\operatorname {a0} \right )^{\frac {\operatorname {a0} \operatorname {b1} \operatorname {c2} -\operatorname {a0} \operatorname {b2} \operatorname {c1} -\operatorname {a1} \operatorname {b0} \operatorname {c2} +\operatorname {c0} \operatorname {b2} \operatorname {a1}}{\operatorname {a1}^{2} \operatorname {b2}}} {\mathrm e}^{\frac {\operatorname {c2} \left (\operatorname {b2} y +\operatorname {b0} \right ) \operatorname {a1}^{2}+\left (\left (\operatorname {c1} x -\operatorname {c2} y \right ) \operatorname {b2}^{2}-\operatorname {c2} \left (\operatorname {b1} x +\operatorname {b0} \right ) \operatorname {b2} -\operatorname {a0} \operatorname {b1} \operatorname {c2} \right ) \operatorname {a1} +x \,\operatorname {b2}^{2} \left (\operatorname {b1} \operatorname {c2} -\operatorname {c1} \operatorname {b2} \right )}{\left (\operatorname {a1} -\operatorname {b2} \right ) \operatorname {b2}^{2} \operatorname {a1}}}}{\left (\operatorname {a1} -\operatorname {b2} \right ) \operatorname {b2}}\]
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6.5.1.8 [1203] Problem 8
problem number 1203
Added March 10, 2019.
Problem Chapter 5.2.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y)\)
\[ a y w_x + (b_1 x+b_0) w_y = (c_1 x+c_0) w + s_1 x+s0 \]
Mathematica ✓
ClearAll["Global`*"];
pde = a*y*D[w[x, y], x] + (b1*x + b0)*D[w[x, y], y] == (c1*x + c0)*w[x, y] + s1*x + s0;
sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y], {x, y}], 60*10]];
\begin{align*}& \left \{w(x,y)\to \exp \left (\frac {(\text {b0} \text {c1}-\text {b1} \text {c0}) \text {arctanh}\left (\frac {\text {b0}+\text {b1} x}{\sqrt {\text {b1}} \sqrt {a y^2}}\right )-\sqrt {\text {b1}} \text {c1} \sqrt {a y^2}}{\sqrt {a} \text {b1}^{3/2}}\right ) \left (\int _1^x-\frac {\exp \left (\frac {\frac {\sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[1]^2+2 \text {b0} K[1]} \text {c1}}{\text {b1}}+\frac {(\text {b1} \text {c0}-\text {b0} \text {c1}) \text {arctanh}\left (\frac {\text {b0}+\text {b1} K[1]}{\sqrt {\text {b1}} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[1]^2+2 \text {b0} K[1]}}\right )}{\text {b1}^{3/2}}}{\sqrt {a}}\right ) (\text {s0}+\text {s1} K[1])}{\sqrt {a} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[1]^2+2 \text {b0} K[1]}}dK[1]+c_1\left (-\frac {-a y^2+2 \text {b0} x+\text {b1} x^2}{2 a}\right )\right )\right \}\\& \left \{w(x,y)\to \exp \left (\frac {(\text {b1} \text {c0}-\text {b0} \text {c1}) \text {arctanh}\left (\frac {\text {b0}+\text {b1} x}{\sqrt {\text {b1}} \sqrt {a y^2}}\right )+\sqrt {\text {b1}} \text {c1} \sqrt {a y^2}}{\sqrt {a} \text {b1}^{3/2}}\right ) \left (\int _1^x\frac {\exp \left (-\frac {\frac {\sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[2]^2+2 \text {b0} K[2]} \text {c1}}{\text {b1}}+\frac {(\text {b1} \text {c0}-\text {b0} \text {c1}) \text {arctanh}\left (\frac {\text {b0}+\text {b1} K[2]}{\sqrt {\text {b1}} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[2]^2+2 \text {b0} K[2]}}\right )}{\text {b1}^{3/2}}}{\sqrt {a}}\right ) (\text {s0}+\text {s1} K[2])}{\sqrt {a} \sqrt {-\text {b1} x^2-2 \text {b0} x+a y^2+\text {b1} K[2]^2+2 \text {b0} K[2]}}dK[2]+c_1\left (-\frac {-a y^2+2 \text {b0} x+\text {b1} x^2}{2 a}\right )\right )\right \}\\\end{align*}
Maple ✓
restart;
pde := a*y*diff(w(x,y),x)+ (b1*x+b0)*diff(w(x,y),y) = (c1*x+c0)*w(x,y)+s1*x+s0;
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y))),output='realtime'));
\[w \left (x , y\right ) = {\mathrm e}^{\frac {\operatorname {c1} \sqrt {a^{2} y^{2}}}{a \operatorname {b1}}} {\left (\frac {a \left (\operatorname {b1} x +\operatorname {b0} \right )+\sqrt {a^{2} y^{2}}\, \sqrt {a \operatorname {b1}}}{\sqrt {a \operatorname {b1}}}\right )}^{-\frac {\operatorname {b0} \operatorname {c1} -\operatorname {b1} \operatorname {c0}}{\operatorname {b1} \sqrt {a \operatorname {b1}}}} \left (\int _{}^{x}\frac {\left (\textit {\_a} \operatorname {s1} +\operatorname {s0} \right ) {\left (\frac {\sqrt {a \operatorname {b1}}\, \sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) \operatorname {b1} +\left (2 \textit {\_a} -2 x \right ) \operatorname {b0} +a \,y^{2}\right )}+\left (\textit {\_a} \operatorname {b1} +\operatorname {b0} \right ) a}{\sqrt {a \operatorname {b1}}}\right )}^{\frac {\operatorname {b0} \operatorname {c1} -\operatorname {b1} \operatorname {c0}}{\operatorname {b1} \sqrt {a \operatorname {b1}}}} {\mathrm e}^{-\frac {\operatorname {c1} \sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) \operatorname {b1} +\left (2 \textit {\_a} -2 x \right ) \operatorname {b0} +a \,y^{2}\right )}}{a \operatorname {b1}}}}{\sqrt {a \left (\left (\textit {\_a}^{2}-x^{2}\right ) \operatorname {b1} +\left (2 \textit {\_a} -2 x \right ) \operatorname {b0} +a \,y^{2}\right )}}d \textit {\_a} +f_{1} \left (y^{2}-\frac {\operatorname {b1} \,x^{2}}{a}-\frac {2 \operatorname {b0} x}{a}\right )\right )\]
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