General Solution of Differential Equation

How to Find the General Solution of Differential Equation

General Solution of Differential Equation.
Problems with differential equations are asking you to find an unknown function or functions, rather than a number or set of numbers as you would normally find with an equation like f(x) = x2 + 9.

For example, the differential equation dydx = 10x is asking you to find the derivative of some unknown function y that is equal to 10x.

General Solution of Differential Equation: Example

Example problem #1: Find the general solution for the differential equation dydx = 2x.

Step 1: Use algebra to get the equation into a more familiar form for integration:
dydx = 2x →
dy = 2x dx

Step 2: Integrate both sides of the equation:
∫dy = ∫2x dx →
&int1 dy = &int2x dx →
y = x2 + C

Example problem #2: Find the general solution for the differential equation dydx = x2 – 3

Step 1: Use algebra to get the equation into a more familiar form for integration:
dydx = x2 – 3→
dy = x2 – 3 dx

Step 2: Integrate both sides of the equation:
∫ dy = ∫x2 – 3 dx →
∫ 1 dy = ∫x2 – 3 dx →
y = x33 -3x + C

Sample problem #3: Find the general solution for the differential equation θ2 dθ = sin(t + 0.2) dt.

Step 1: Integrate both sides of the equation:
∫ θ2 dθ = ∫sin(t + 0.2) dt →
θ3 = -cos(t + 0.2) + C
That’s how to find the general solution of differential equations!

Tip: If your differential equation has a constraint, then what you need to find is a particular solution. For example, dydx = 2x ; y(0) = 3 is an initial value problem that requires you to find a solution that satisfies the constraint y(0) = 3.

References

Larson & Edwards. Calculus.


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