Differential Equations

Board: JEE Main |Exam:JEEMain

JEEMainstudynotesforDifferentialEquationsbyAjayYadav.

KeyConcepts

OrderandDegree

Order=highestderivative.Degree=powerofhighestderivative.LinearDE:coefficientsdependonlyonx.

VariableSeparable

dy/dx=f(x)g(y)→dy/g(y)=f(x)dx.

HomogeneousDE

dy/dx=f(y/x).Substitutey=vx.

ExactDE

Mdx+Ndy=0where∂M/∂y=∂N/∂x.Solution:∫Mdx+∫Ndy=C.

LinearDE

dy/dx+Py=Q.IF=e^∫Pdx.y(IF)=∫Q(IF)dx+C.

BernoulliEquation

dy/dx+Py=Qyⁿ.Substitutev=y¹⁻ⁿtomakelinear.

ImportantFormulas

VS	dy/g(y)=f(x)dx
Homogeneous	y=vx,dy/dx=v+x(dv/dx)
LinearIF	IF=e∫Pdx
Bernoulli	v=y¹⁻ⁿ

SolvedExamples

Example1:Solvedy/dx=xy.

Solution:dy/y=xdx⇒ln|y|=x²/2+C. y=Cex²/².

Practice Questions

1. Solve dy/dx=(x+y)/x.
2. Solve dy/dx+x/y=0.
3. Solve dy/dx+2xy=x.
4. Find order/degree of (d²y/dx²)³+(dy/dx)⁴=0.
5. Solve dy/dx + ycotx = 2cosx.

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