# Manual Homogeneous linear substitutions

Recall that a function is homogeneous if:. Differentiating the substitution, using the product rule to differentiate the right-hand side, gives:. A cancellation such as this will always happen and the remaining equation should be solvable by a simple method such as separation or an integrating factor. The above equation is solvable by separation.

### Homogeneous Functions

The equation can now be separated and integrated directly using standard integrals :. To do this, differentiate the given substitution:. The chain rule is used to differentiate the right-hand side:. Explanation : To find the general solution to the given system first find the eigenvalues and eigenvectors. Report an Error.

solve differential equation with substitution

Example Question 1 : Homogeneous Linear Systems. Explanation : To solve the homogeneous system, we will need a fundamental matrix.

## Homogeneous Differential Equation: Functions, Videos, Solved Examples

Explanation : So this is a homogenous, second order differential equation. Solve the second order differential equation: Subject to the initial values:.

Solve the differential equation for y: Subject to the initial condition:. Explanation : So this is a homogenous, first order differential equation. Solve the third order differential equation:. Explanation : So this is a homogenous, third order differential equation. Solve the differential equation: Subject to the initial conditions:.

Explanation : A saddle point phase plane results from two real eigenvalues of different signs.

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Example Question 71 : Differential Equations. Find the general solution to the system of ordinary differential equations where. Copyright Notice.

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