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- 1. EXACT EQUATIONS Non-linear, Non-seperable first order differential equation
- 2. Techniques we know: 2 Differential equation Type Solving method M(y)y'=n(x) Non-linear seperable Seperation of variables y'+P(x)y=Q(x) Linear Integrating factor M(x,y)+N(x,y)y'=0 Non- seperable non-linear ?
- 3. IMPLICIT DIFFERENTIATION Let us take an example function: f(x,y)= =>f(x,y)'=( )' = Now differentiate the function partially: By comaring above equations we get:
- 4. 4 So ,this is the general rule: =M(x,y)+N(x,y)y' =>To solve problems of the form: M(x,y)+N(x,y)y'=0 What we do? We should find: f(x,y)
- 5. Solving nonlinear non- separable equations • To find f(x,y):
- 6. EXAMPLE: • Let the differential equation be : • • => commonpart • & • Common part= • h(x)= • g(y)= =>
- 7. If we try to find the general solution of 3y+2xy'=0 where if we do but f(x,y)=h(x)+commonpart+g(y) but the above integrations doesn't have commonpart or h(x) and g(y). To find G.S for the above equation we reduce it to exact equation. let us know what is exact equation..
- 8. Let's know what is exact equation ? • Remember , order doesn't matter for partial derivatives: => • • => • & • => • • => To solve the differential of the form M(x,y)+N(x,y)y'=0 the necessary condition is • => and the above equation is called exact.
- 9. To solve problems of the form M(x,y)+N(x,y)y'=0 Step 1:- Observe Step 2:-Compute Mdx + Ndy Step 3:- write the general solution as f(x,y)= M(x,y)dx+ N(x,y)dy If a differential equation Mdx+Ndy=0 which is not exact can be made exact by multiplying it with a suitable µ(x,y)≠0 then µ(x,y) is called an integrating factor of the equation Mdx+Ndy=0. 9
- 10. Differential equation Type Method of solving M(y)y'=n(x) Non-linear separable Seperation of variables y'+P(x)y=Q(x) Linear Integrating factor M(x,y)+N(x,y)y'=0 Non-linear Non separable Exact Method of exact equations SUMMARY