Did I solve this diff eq substition problem properly?

  • Context: Graduate 
  • Thread starter Thread starter utnip123
  • Start date Start date
  • Tags Tags
    Diff eq
Click For Summary
SUMMARY

The forum discussion centers on solving the differential equation xy' = yln(xy) using substitution. The user successfully applies the substitution v = ln(xy), leading to the integral ∫(dv/(v+1)) = ∫(dx/x). The solution results in ln(v+1) = ln(x) + C, which simplifies to v + 1 = Cx, ultimately yielding ln(xy) + 1 = Cx. The user suggests verifying the solution by solving for y and checking against the original equation.

PREREQUISITES
  • Understanding of differential equations and their solutions
  • Familiarity with substitution methods in calculus
  • Knowledge of logarithmic properties and integrals
  • Ability to manipulate and solve equations involving natural logarithms
NEXT STEPS
  • Explore advanced techniques in solving differential equations
  • Learn about the method of integrating factors for differential equations
  • Study the applications of logarithmic differentiation in solving equations
  • Investigate numerical methods for verifying solutions to differential equations
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus and differential equations, as well as educators seeking to enhance their understanding of substitution methods in solving complex equations.

utnip123
Messages
9
Reaction score
0
[itex]xy'=yln(xy)[/itex]
[itex]xdy=yln(xy)dx[/itex]
[itex]\frac{dy}{y}[/itex]=[itex]\frac{ln(xy)dx}{x}[/itex]

Substitution:

[itex]v=ln(xy)[/itex]
[itex]dv[/itex] = [itex]\frac{dy}{y}[/itex]-[itex]\frac{dx}{x}[/itex]
[itex]dv[/itex]-[itex]\frac{dx}{x}[/itex]=[itex]\frac{vdx}{x}[/itex]

∫[itex]\frac{dv}{v+1}[/itex]=∫[itex]\frac{dx}{x}[/itex]

[itex]ln(v+1)=ln x + C[/itex]

v+1 = Cx
ln(xy) +1 = Cx

Would that basically be the complete answer?
 
Physics news on Phys.org
It might be interesting to solve this last equation for y=... and verify the answer with the initial equation.
 

Similar threads

Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
High School How Can We Model the Coronavirus Using Predator-Prey Dynamics?
  • · Replies 18 ·
Replies
18
Views
3K
Undergrad Analyzing a 2nd Order Non-Linear ODE with Variable Substitution
  • · Replies 2 ·
Replies
2
Views
2K
MHB Can You Solve This Initial Value Problem and Determine Its Solution Interval?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Solve Ordinary Differential Equation: ln(y)=xy
  • · Replies 11 ·
Replies
11
Views
3K
MHB Solving Homogeneous ODE: $\displaystyle x(y-3x)\frac{dy}{dx}=2y^2-9xy+8x^2$
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why Lagrange’s method of solving Pp + Qq=R works?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Solving 2D Heat Equation w/ FEM & Galerkin Method
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad I with differential equation(3x^2-2y^2)(dy/dx)=2xy
  • · Replies 9 ·
Replies
9
Views
9K
MHB Solve DE: Reviewing Problems Before Classes Start
  • · Replies 3 ·
Replies
3
Views
2K