# SECTION 2.2 # Example 1 LHS := Int(y^2,y); NiM+JSRMSFNHLSUkSW50RzYkKiQpJSJ5RyIiIyIiIkYq LHS := value(%); NiM+JSRMSFNHLCQqJiIiJCEiIiUieUdGJyIiIg== RHS := Int(x-5,x); NiM+JSRSSFNHLSUkSW50RzYkLCYlInhHIiIiIiImISIiRik= RHS := value(%); NiM+JSRSSFNHLCYqJiMiIiIiIiNGKCokKSUieEdGKUYoRihGKComIiImRihGLEYoISIi sol := LHS = RHS + C; NiM+JSRzb2xHLywkKiYiIiQhIiIlInlHRigiIiIsKComI0YrIiIjRisqJCklInhHRi9GK0YrRisqJiIiJkYrRjJGK0YpJSJDR0Yr solve(sol,y); NiUsJComIiIjISIiLCgqJiIjNyIiIiklInhHRiVGKkYqKiYiJD8iRipGLEYqRiYqJiIjQ0YqJSJDR0YqRiojRioiIiRGKiwmKiYiIiVGJkYnRjJGJiooXiMjRipGNkYqRjMjRipGJSlGJ0YyRipGKiwmKiZGNkYmRidGMkYmKiheIyNGJkY2RipGM0Y6RjtGKkYq ysol := %[1]; NiM+JSV5c29sRywkKiYiIiMhIiIsKComIiM3IiIiKSUieEdGJ0YsRiwqJiIkPyJGLEYuRixGKComIiNDRiwlIkNHRixGLCNGLCIiJEYs # Plot Figure 2.3 with(plots): ?contourplot eq := 3*(LHS - RHS); NiM+JSNlcUcsKCokKSUieUciIiQiIiJGKiomI0YpIiIjRioqJCklInhHRi1GKkYqISIiKiYiIzpGKkYwRipGKg== contourplot(eq,x=-3..15,y = -6..6,contours = [-24,-12,0,12,24]); 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# Alternative solution using dsolve: y := 'y'; NiM+JSJ5R0Yk de := diff(y(x),x) = (x-5)/y(x)^2; NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJiwmRiwiIiIiIiYhIiJGL0YpISIj dsolve(de); NiUvLSUieUc2IyUieEcsJComIiIjISIiLCgqJiIjNyIiIilGJ0YqRi9GLyomIiQ/IkYvRidGL0YrKiYiIilGLyUkX0MxR0YvRi8jRi8iIiRGLy9GJCwmKiYiIiVGK0YsRjZGKyooXiMjRitGO0YvRjcjRi9GKilGLEY2Ri9GLy9GJCwmKiZGO0YrRixGNkYrKiheIyNGL0Y7Ri9GN0Y/RkBGL0Yv # Comparing this result with the first one, it appears dsolve uses a similar soltion technique
in this case. # Example 2 LHS := Int(1/(y-1),y); NiM+JSRMSFNHLSUkSW50RzYkKiYiIiJGKSwmJSJ5R0YpRikhIiJGLEYr LHS := value(%); NiM+JSRMSFNHLSUjbG5HNiMsJiUieUciIiJGKiEiIg== LHS := ln(abs(y-1)); #Maple omitted the absolute value sign above, the correct result is: NiM+JSRMSFNHLSUjbG5HNiMtJSRhYnNHNiMsJiUieUciIiJGLSEiIg== RHS := Int(1/(x+3),x); NiM+JSRSSFNHLSUkSW50RzYkKiYiIiJGKSwmJSJ4R0YpIiIkRikhIiJGKw== RHS := value(%); NiM+JSRSSFNHLSUjbG5HNiMsJiUieEciIiIiIiRGKg== RHS := ln(abs(x+3)); NiM+JSRSSFNHLSUjbG5HNiMtJSRhYnNHNiMsJiUieEciIiIiIiRGLQ== sol := LHS = RHS + C; NiM+JSRzb2xHLy0lI2xuRzYjLSUkYWJzRzYjLCYlInlHIiIiRi4hIiIsJi1GJzYjLUYqNiMsJiUieEdGLiIiJEYuRi4lIkNHRi4= Following the second approach: subs({x=-1,y=0},sol); NiMvLSUjbG5HNiMtJSRhYnNHNiMhIiIsJi1GJTYjLUYoNiMiIiMiIiIlIkNHRjE= solve(%,C); NiMsJC0lI2xuRzYjIiIjISIi subs(C=-ln(2),sol); NiMvLSUjbG5HNiMtJSRhYnNHNiMsJiUieUciIiJGLCEiIiwmLUYlNiMtRig2IywmJSJ4R0YsIiIkRixGLC1GJTYjIiIjRi0= solve(%,y) assuming y-1 < 0 , x+3 >0; NiMsJiomIiIjISIiJSJ4RyIiIkYmI0YoRiVGJg== # Alternative solution using dsolve: y := 'y'; de := diff(y(x),x) = (y(x)-1)/(x+3); NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJiwmRikiIiJGLyEiIkYvLCZGLEYvIiIkRi9GMA== dsolve({de,y(-1)=0}); NiMvLSUieUc2IyUieEcsJiMiIiIiIiMhIiIqJkYrRixGJ0YqRiw= # Example 3 LHS := Int(cos(y)+exp(y),y); NiM+JSRMSFNHLSUkSW50RzYkLCYtJSRjb3NHNiMlInlHIiIiLSUkZXhwR0YrRi1GLA== LHS := value(%); NiM+JSRMSFNHLCYtJSRzaW5HNiMlInlHIiIiLSUkZXhwR0YoRio= RHS := Int(6*x^5 - 2*x + 1,x); NiM+JSRSSFNHLSUkSW50RzYkLCgqJiIiJyIiIiklInhHIiImRitGKyomIiIjRitGLUYrISIiRitGK0Yt RHS := value(%); NiM+JSRSSFNHLCgqJCklInhHIiInIiIiRioqJClGKCIiI0YqISIiRihGKg== sol := LHS = RHS + C; NiM+JSRzb2xHLywmLSUkc2luRzYjJSJ5RyIiIi0lJGV4cEdGKUYrLCoqJCklInhHIiInRitGKyokKUYxIiIjRishIiJGMUYrJSJDR0Yr # Alternative solution using dsolve: y := 'y'; de := diff(y(x),x) = (6*x^5 - 2*x + 1)/(cos(y(x)) + exp(y(x))); NiM+JSJ5R0Yk NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJiwoKiYiIiciIiIpRiwiIiZGMUYxKiYiIiNGMUYsRjEhIiJGMUYxRjEsJi0lJGNvc0c2I0YpRjEtJSRleHBHRjpGMUY2 dsolve(de); NiMvLC4qJCklInhHIiInIiIiRikqJClGJyIiI0YpISIiRidGKS0lJHNpbkc2Iy0lInlHNiNGJ0YtLSUkZXhwR0YwRi0lJF9DMUdGKSIiIQ== # Notice that, unlike with Examples 1 and 2, dsolve does not attempt to give an explicit # solution in this case.