Substituting \(tx\) for \(x\) and \(ty\) for \(y\) in \(M(x,y)\) we get \[M(tx,ty)=(tx)^2+(tx)(ty)=t^2(x^2+xy)= t^2M(x,y)\nonumber\] and \(M(x,y)\) is a homogeneous function of degree 2. Substituting ...Substituting \(tx\) for \(x\) and \(ty\) for \(y\) in \(M(x,y)\) we get \[M(tx,ty)=(tx)^2+(tx)(ty)=t^2(x^2+xy)= t^2M(x,y)\nonumber\] and \(M(x,y)\) is a homogeneous function of degree 2. Substituting \(tx\) for \(x\) and \(ty\) for \(y\) in \(N(x,y)\) we get \[N(tx,ty)=(ty)^2=t^2y^2= t^2N(x,y)\nonumber\] and \(N(x,y)\) is a homogeneous function of degree 2.
There are several nonlinear first order equations whose solution can be obtained using special techniques. We conclude this chapter by looking at a few of these equations named after famous mathematic...There are several nonlinear first order equations whose solution can be obtained using special techniques. We conclude this chapter by looking at a few of these equations named after famous mathematicians of the 17−18th century inspired by various applications
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
This page covers methods for simplifying non-separable differential equations through substitution, particularly Bernoulli equations, highlighting the significance of making effective substitutions to...This page covers methods for simplifying non-separable differential equations through substitution, particularly Bernoulli equations, highlighting the significance of making effective substitutions to achieve linear forms for easier solutions. It provides examples demonstrating the use of the substitution \(v = \frac{y}{x}\) for homogeneous equations, illustrating the process of transforming complex equations into simpler, separable ones.
This section deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.
